Approximation-Free Prescribed Performance Control

Transcription

1 Preprints of the 8th IFAC World Congress Milano Italy August 28 - September 2 2 Approximation-Free Prescribed Performance Control Charalampos P. Bechlioulis and George A. Rovithakis Department of Electrical & Computer Engineering Aristotle University of Thessaloniki Thessaloniki 5424 Greece chmpechl@auth.gr and robi@eng.auth.gr. Abstract: A universal controller is designed for strict feedback systems with unknown nonlinearities capable of guaranteeing for any a priori known initial state condition bounded signals in the closed loop as well as prescribed performance for the output tracking error. By prescribed performance we mean that the output tracking error converges to a predefined arbitrarily small residual set with convergence rate no less than a certain prespecified value exhibiting maximum overshoot less than some sufficiently small preassigned constant. The proposed control scheme is of low complexity and requires reduced levels of a priori system knowledge. A comparative simulation study verifies the superiority of the proposed control scheme against the well known backstepping technique which has been the main tool for designing controllers for the class of systems considered. Keywords: Prescribed performance control backstepping strict feedback systems.. INTRODUCTION Consider an n-th order strict feedback system described as follows: ẋ i = f i x i +g i x i x i+ i =...n ẋ n = f n x n +g n x n u y = x where x i R i =...n are the states which are available for measurement with initial conditions x i i =...n while x i = [x x i ] T. Moreover u R is the control input y R is the output and the functions f i : R i R g i : R i R i =...n are smooth. The functions g i i =...n which are referred to as control coefficients are also either strictly positive or strictly negative. Such a property guarantees that system is uniformly strongly controllable. The control objective is to design a control input u such that the output y tracks a desired trajectory y d t which is a smooth bounded function of time with bounded derivatives. At the early 9s a recursive design procedure called backstepping Krstic et al. 995 emerged in an attempt to provide a control solution for the tracking problem of system. With this powerful methodology the construction of both feedback control laws and associated Lyapunov functions became systematic. Strong properties of global stability and tracking were built onto the nonlinear system the system nonlinearities were considered completely known in a number of steps equal to the system order. Yet the true potential of backstepping was discovered only when this approach was developed for strict feedback systems with uncertainties. Adaptive This work was supported in part by the Alexander S. Onassis Public Benefit Foundation under Grant G ZD 45 / backstepping Krstic et al. 995 achieved the control objective in the presence of unknown parameters while robust neuro/fuzzy adaptive backstepping Spooner et al. 22 Ge et al. 22 Farrell and Polycarpou 26 dealt with model uncertainties. The structured way in which the backstepping approach dealt unknown parameters and model uncertainties contributed to its instant popularity and rapid acceptance. However although the backstepping technique has been well established in the current literature certain issues still remain open. An inherent drawback of backstepping is its complexity owing to the coupling between the design steps. The problem becomes harder as the system order increases thus making implementation difficult. Another central issue encountered when system possesses uncertainties is the so called loss of controllability problem. Although the actual system is assumed controllable that is g i x i > i =...n the estimation model may lose its controllability at some points in time when at least one of the estimations ĝ i x i ˆθ i =...n owing to parameter ˆθ adaptation. Moreover in case of model uncertainties where approximation structures i.e. neural networks fuzzy systems etc. are utilized it is well known that the results are valid only inside a compact set where the approximation holds. However determining a priori the aforementioned compact set and ensuring that systemtrajectoriesareconfinedwithinitaretwoopenand challenging problems. Typically this problem is bypassed when initializing from a subset of the aforementioned set with sufficiently small initial estimate errors and for an appropriate selection of the control gains. However although the existence of such initialization/selection process is assured no construction methodology is provided other than a trial and error procedure. Another important Copyright by the International Federation of Automatic Control IFAC 26

2 Preprints of the 8th IFAC World Congress Milano Italy August 28 - September 2 2 issue associated with the tracking problem of system concerns tracking error performance. When the system nonlinearities are completely known or involve parametric uncertainties backstepping achieves asymptotic tracking. The presence of model uncertainties increases significantly the complexity of the problem. Convergence of the tracking error to a residual set whose size depends on some control gains and bounded though unknown terms can be achieved. However no systematic procedure exists to accurately compute the required upper bounds thus making the a priori selection of the aforementioned control gains to satisfy certain steady state behavior practically impossible. Moreover performance issues on transient behavior such as convergence rate and maximum overshoot are difficult to be established analytically even for the case of known nonlinearities. Recently Bechlioulis and Rovithakis 29 have proposed a neuro adaptive controller for system capable of guaranteeing prescribed performance on the tracking error yt y d t. By prescribed performance it is meant that the tracking error converges to a predefined arbitrarily small residual set with convergence rate no less than a certain prespecified value exhibiting maximum overshoot less than some sufficiently small preassigned constant. However the aforementioned inherent restrictions related to design complexity and utilizing approximation based techniques where not overcome. In this paper we establish a general framework referred to as Prescribed Performance Control PPC that systematically converts the prescribed performance tracking problem for into a robust stabilization problem for anothersystemwhichretainsthecontrollabilityofand involves both the nonlinearities of the original system as wellasthedesiredperformancecharacteristics.itisproven that only the boundedness of the aforementioned system s states is sufficient to solve the prescribed performance trackingproblemfortheoriginalsystem.consequently an approximation free controller is designed to guarantee the required boundedness properties. The main contributions of this work can be summarized as follows: PPC achieves global results in the sense that given any a priori known initial state x n and any output performance requirements prescribed performance tracking is achieved. Apart from a sufficient controllability condition for no extra assumptions are made and no extra information is required concerning its nonlinearities. Furthermore only the desired trajectory y d t and none of its higher order derivatives is utilized. The proposed controller is approximation-free and significantly less complex compared to the backstepping approach which is the main tool for designing controllers for. Moreover contrary to what is the common practice in the relevant literature system performance is isolated from control gains selection thus leading to further simplification of the control design procedure.. Definitions and Preliminaries At this point we recall some definitions and preliminary results which are necessary in the subsequent analysis. Definition. Ge et al. 24 A smooth function N ζ is called Nussbaum function if it is equipped with the following properties: lim t sup t lim t inf t t t N ζdζ = + N ζdζ =. Physically it can be visualized as a function of infinite gain and infinite switching frequency. An even Nussbaum function is N ζ = exp aζ 2 coswζ where aw >. Lemma. Ge et al. 24 Let Vt and ζt be some smooth functions defined on [t f with V t t [t f and N ζ be an even Nussbaum-type function. If the following inequality holds: V t e c t t gxτn ζ+ ζe c τ dτ +c where c c are some positive constants and gxτ is a function taking values in the compact set L = [l l + ] with / L then V t and ζt are bounded on [t f. Definition 2. Bechlioulis and Rovithakis 28 A smooth bounded function ρ : R + R + will be called a performance function if ρt is decreasing and lim t ρt = ρ >. As it is stated in Bechlioulis and Rovithakis 28 and Bechlioulis and Rovithakis 29 prescribed transient and steady state bounds for the output tracking error e t = yt y d t can be satisfied by guaranteeing: Mρ t < e t < ρ t in case e 2 ρ t < e t < Mρ t in case e for all t where M and ρ t a performance function see Definition 2 associated with e t. As 2 implies we employ only one set of the performance bounds and specifically the one associated with the sign of e. The aforementioned statements are clearly illustrated in Fig. for an exponentially decaying performance function of the form ρ t = ρ ρ exp lt + ρ. The constant ρ represents the maximum allowable size of the tracking error e t at the steady state which can be set arbitrarily small to a value reflecting the resolution of the measurement device thus achieving practical convergence of e t to zero. Furthermore the decreasing rate of ρ t introduces a lower bound on the required speed of convergence of e t while the maximum overshoot is prescribed less than Mρ which may even become zero by setting M =. Thus the appropriate selection of the performance function ρ t as well as of the design constant M imposes specific performance bounds on the system output trajectory. 2. PRESCRIBED PERFORMANCE CONTROL In this section we shall first derive a sufficient condition for the solution of the prescribed performance tracking problem of system. Specifically we shall demonstrate thatachievingboundednessofthestatesofanovelsystem involving both the nonlinearities of as well as the desiredperformancecharacteristicsviasmoothandbounded control signals is sufficient to solve the stated problem. 27

3 Preprints of the 8th IFAC World Congress Milano Italy August 28 - September 2 2 e t ρ ρ e t ρ t Mρ t Mρ t ρ t a b tsec Fig..Graphicalrepresentationof2incaseae and in case b e. Consequently a significantly less complex controller when compared to the backstepping design will be presented. 2. A Sufficient Condition Let us consider system and any a priori known initial condition x n. For the output tracking error e t = x t y d t we associate a performance function ρ t and a constant M to guarantee: a the successful prescription of the output error performance requirements see subsection. and b the satisfaction of ρ > e. We further define the state errors e i t = x i t a i t i = 2...n where a i t i =...n are some continuously differentiable intermediate control signals to be designed and ρ i t i = 2...n are some designer specified performance functions satisfying ρ i > e i i = 2...n. Moreover let S i i =...n be some smooth strictly increasing functions defining onto mappings: S : MM 3 S i : i = 2...n. where: MM = { M in case e M in case e. For example candidate functions could be S M exp +M exp exp +exp and S i = tanh i = 2...n. The following theorem provides a sufficient condition for the solution of the prescribed performance tracking problem for system. Theorem. Consider the system: ε i = f ρ i t ds i i v i +g i v i a i t dε ε i i +g i v i ρ i+ ts i+ ε i+ ȧ i t ρ i ts i ε i i =...n 4 ε n = f ρ nt ds n n v dεn εn n +g n v n u ȧ n t ρ n ts n ε n where ε i R i =...n are the states with initial conditions ε i = S i i =...n and v i ei ρ i [v v i ] T where v i = ρ i ts i ε i +a i t i =...n 5 witha t = y d t.ifthecontrolsignalsa i ti =...n with a n t = ut are designed such that ε i t a i t L i =...n then the prescribed performance tracking problem for is solved. Proof. Notice that the initial conditions of 4 i.e. ε i i =...n are finite owing to 3 as well as to ρ i > e i i =...n.moreoversupposethatε i t a i t L i =...n for some appropriately selected control signals a i t i =...n with a n t = ut. Consequently ρ i t ds i ε i > i =...n for all t. Thus 4 yields: ρ i t ds i ε i ε i + ρ i ts i ε i +ȧ i t = f i v i +g i v i ρ i+ ts i+ ε i+ +a i t i =...n ρ n t dsn dε n ε n ε n + ρ n ts n ε n +ȧ n t = f n v n +g n v n u. Differentiating 5 with respect to time we obtain: v i = ρ i t ds i ε i ε i + ρ i ts i ε i +ȧ i t i =...n. Hence 6 becomes: v i = f i v i +g i v i v i+ i =...n v n = f n v n +g n v n u. 7 The initial conditions of 4 satisfy ε i = S i ei ρ i i =...n which together with 5 at t = lead to v i = x i i =...n. Therefore by comparing and 7 we conclude v i t = x i t i =...n for all t. Furthermore owing to 3 5 and the fact that ε i t L i =...n we arrive at: Mρ t < e t < ρ t in case e ρ t < e t < Mρ t in case e t 8 e i t < ρ i t i = 2...n t. 9 Accordingto Subsection.8 isthemathematical interpretation of prescribed performance for e t. Moreover owing to 8 9 and the boundedness of y d t a i t i =...n we conclude the boundedness of x n which completes the proof. Remark. System 4 retains the controllability of the original system owing to the fact that i = ρ it ds i dε ε i i...n are strictly positive by construction. As a consequence for this class of systems any set of prescribed performance bounds for the output tracking error can be achieved. Notice also that system 4 is used only for analysis. It cannot be implemented as it involves the unknown nonlinear functions f i g i i =...n. However the signals ε i t i =...n which will be utilized in a i t i =...n can be calculated owing to 5 and the fact that v i t = x i t i =...n for all t via: ε i t = S i 2.2 PPC Design xi t a i t ρ i t i =...n t. 6 Following Theorem to solve the prescribed performance tracking problem for we present in this subsection an 28

4 Preprints of the 8th IFAC World Congress Milano Italy August 28 - September 2 2 approximation-free design methodology based on system 4 that leads to a low-complexity controller capable of guaranteeing the boundedness of ε i t i =...n as well as of all other signals in the closed loop. Theorem 2. Consider system a smooth bounded desired trajectory y d t the output performance requirements introduced via M ρ t and the performance functions ρ i t i = 2...n. The control signals: a i = N ζ i η i ε i η i ζ i = ρ i t ds i ε i η i = k i ρ i t dsi ε i +ρ it ds i ε i ε i i =...n with N an even Nussbaum function k i > ρ i > x i a i i =...n and ut = a n t where ε i t i =...n are given by: ε i t = S i xi t a i t ρ i t i =...n for some smooth strictly increasing functions S i i =...n satisfying 3 solve the prescribed performance tracking problem for. Proof. The proof is divided into n-steps. Step : Consider the first state of 4: ε = ρ t ds dε ε f v +g v a t +g v ρ 2 ts 2 ε 2 ȧ t ρ ts ε. 2 Let us also consider the positive definite and radially unbounded function V = 2 ε2 whose time derivative along 2 is given by: ε V = f ρ t ds dε ε v +g v a t +g v ρ 2 ts 2 ε 2 ȧ t ρ ts ε. 3 Substituting for i = into 3 and after adding and subtracting ζ we obtain: V = k ε 2 +g v N ζ + ζ where k ε ρ t ds dε ε 2 + ε ρ t ds dε ε F ε ε 2 v t F ε ε 2 v t = f v +g v ρ 2 ts 2 ε 2 ȧ t ρ ts ε. Notice that v as defined in 5 is bounded by construction for all ε R and t i.e. there exists a positive constant v such that v v t. Owing to the smoothness of f and g as well as to v v t applying the Extreme Value Theorem we conclude the existence of some positive constants f ḡ such that f v f and g v ḡ t. Moreover ρ 2 ts 2 ε 2 ρ ts ε as well as ȧ t = ẏ d t are also bounded for all ε R ε 2 R and t. Thus there exists a positive constant F such that F ε ε 2 v t F t. Hence completing the squares we get: V k ε 2 +g v N ζ + ζ + F 2 4k. 4 Multiplying 4 by e 2kt and integrating we arrive at: V t e 2k t t g v N ζ + ζ e 2kτ dτ +V + F 2. 8k 2 5 Invoking the fact that g is bounded away from zero a sufficient controllability condition we also obtain that < g g v ḡ t for a positive constant g. Thus applying Lemma we conclude the boundedness of V t ζ t and consequently the boundedness of ε t η t a t ε t and ζ t for all t [t f. Moreover according to Ryan 99 the continuity of the right hand side of 2 together with the boundedness of its solution for all t [t f guarantee that t f =. Finally differentiating a with respect to time yields: ȧ = dn ζ η ζ η +N ζ ε + η dζ ε t from which owing to the smoothness of the Nussbaum function N as well as of η we also conclude that ȧ t L. Step i 2 i n: We have proven in Step that a t L. Similarly to Step it is straightforward to show that for the second state of 4 there exist positive constants f 2 g 2 ḡ 2 such that f 2 v 2 f 2 and < g 2 g 2 v 2 ḡ 2 t. Following the line of proof of the first step with the positivedefiniteandradiallyunboundedfunctionv 2 = 2 ε2 2 and selecting the intermediate control signal a 2 as in for i = 2 we obtain: V 2 t e 2k 2t t g 2 v 2 N ζ 2 + ζ 2 e 2k 2τ dτ +V 2 + F 2 2 8k 2 2 where the positive constant F 2 satisfies: f 2 v 2 +g 2 v 2 ρ 3 ts 3 ε 3 ȧ t ρ 2 ts 2 ε 2 F 2 for all t owing to the boundedness of f 2 v 2 g 2 v 2 ρ 3 ts 3 ε 3 ȧ t and ρ 2 ts 2 ε 2. Thus we conclude that ε 2 t a 2 t ȧ 2 t L. Applying recursively for all the remaining steps the control signals for i = 3...n we conclude that ε i t a i t L i =...n. Hence according to Theorem we have solved the prescribed performance tracking problem for system. Remark 2. The controller summarized in Theorem 2 achieves global prescribed performance results i.e. given any a priori known initial state x n and any performance requirements for the output the control objective is satisfied without requesting knowledge of system nonlinearities. Further no approximation based schemes i.e. neural networks fuzzy systems etc. have been employed to acquire such knowledge. Moreover compared with the backstepping approach which is the main tool for designing controllers for the proposed methodology proves significantly less complex. The aforementioned attributes are mainly owing to: a at each state of system 4 the unknown nonlinearities involved appear bounded and b we only seek for the boundedness of the states of 4 as Theorem dictates. Moreover since at each step the control coefficients of 4 i.e. g i x i i =...n are bounded Lemma can be easily applied. Remark 3. At each design step the derivative of the intermediate control signal is proven bounded. Hence there is no need for compensation in the subsequent step. In 29

5 Preprints of the 8th IFAC World Congress Milano Italy August 28 - September 2 2 thatrespectthedesignstepsoftheproposedmethodology appear decoupled the coupling issue is the main source of complexity in the backstepping design procedure. Furthermore the proposed scheme is also independent of the timederivativesofy d.certainlythefirstintermediatecontrol signal a depends on y d. However ȧ which involves ẏ d is proven bounded and therefore we do not compensate for ȧ when designing the second intermediate control signal a 2. Through repetition the same holds for all a i i = 3...n thus isolating the appearance of ẏ d or higher order derivatives of y d to subsequent steps. Remark 4. To solve the prescribed performance tracking problem for despite the presence of uncertainties it is sufficient to seek for the boundedness of the states of 4. In addition the actual output tracking error performance is solely determined by ρ t and M. Therefore and contrary to what is the common practice in the relevant literature the selection of the control design parameters k i i =...n is significantly simplified to adopting those values that lead to reasonable control effort. The same reasoning applies to the selection of ρ i t i = 2...n for which according to Theorem the only constraint is the satisfaction of ρ i > e i i = 2...n. Remark 5. In case the signs of g i i =...n are known it can be easily verified that the PPC scheme is further simplified to: a i = signg i η i η i = k i +ρ ρ it ds i it ds i dε ε i i ε i ε i 3. SIMULATION RESULTS i =...n. In this section we shall present a comparative simulation study between the backstepping technique and the PPC method. The simulations will be conducted on the following 3rd order strict feedback system: ẋ = f x +g x x 2 x =.2 ẋ 2 = f 2 x 2 +g 2 x 2 x 3 x 2 =.5 ẋ 3 = f 3 x 3 +g 3 x 3 u x 3 =.5 y = x 6 where f x = x sinx g x =.2 + x 2 f 2 x 2 = x 2 e.5x g 2 x 2 = +x 2 2 f 3 x 3 = x x 2 x 3 g 3 x 3 = 2+ cosx x 2. It can be easily verified that g i x i > i = 23. Further the desired trajectory is selected as y d t = sint. First we shall design a backstepping controller following Krstic et al. 995 considering the system nonlinearities completely known. In that respect we select the control signals as: e = x y d a = g x f x ẏ d +k e e 2 = x 2 a a 2 = g 2 x 2 f 2 x 2 ȧ +g x e +k 2 e 2 e 3 = x 3 a 2 u = g 3 x 3 f 3 x 3 ȧ 2 +g 2 x 2 e 2 +k 3 e 3 7 et x tsec Fig. 2. Tracking error response. The black solid line indicates the tracking error response of the PPC scheme. The grey solid line indicates the tracking error response of the backstepping approach. The black dashed lines indicate the performance bounds. The subplot gives details at the steady state. Notice that both ȧ and ȧ 2 which also involves ä are required to formulate the control input 7. The derivation of the aforementioned signals is very arduous as they involve the system nonlinearities. Therefore as the system order increases their analytic determination becomes harder. Additionally the time derivatives of the desired trajectory up to 3rd order are also required. To continue following the PPC design we arrive at: ε = S x y d ρ t ζ ε η = ρ t ds dε ε η = k ρ t ds dε ε +ρ t ds ε ε dε a = N ζ η ε 2 = S2 x2 a ρ 2 t ζ ε 2 η 2 2 = ρ 2 t ds 2 dε 2 ε 2 η 2 = k 2 ρ 2 t ds2 dε 2 ε 2 +ρ 2t ds 2 ε 2 ε 2 dε 2 a 2 = N ζ 2 η 2 ε 3 = S3 x3 a 2 ρ 3 t ζ ε 3 η 3 3 = ρ 3 t ds 3 dε 3 ε 3 η 3 = k 3 ρ 3 t ds 3 dε 3 ε 3 +ρ 3t ds 3 ε 3 ε 3 dε 3 u = N ζ 3 η 3 8 where N ζ = e.ζ2 cos π 4 ζ S ε = exp2ε exp2ε + S 2 ε 2 = tanhε 2 S 3 ε 3 = tanhε 3 ρ t = 2 e 3 exp 5t + 3 ρ 2 t = 2 e 2. exp 5t +. ρ 3 t = 2 e 3.exp 5t +. the property e i < ρ i i = 23 is satisfied. Notice that no information regarding the system nonlinearities is utilized in 8. Moreover comparing to the backstepping case all signals required to produce 8 can 3

6 Preprints of the 8th IFAC World Congress Milano Italy August 28 - September 2 2 ut In this paper we established a general framework to handle the prescribed performance tracking problem for strict feedback systems. After obtaining a novel system that involves both the nonlinearities of the original system as well as the performance requirements we proved that the boundedness of its states is sufficient to solve the prescribed performance tracking problem for the original system. Consequently a state feedback controller was designed to achieve boundedness of its states. The resulted PPC scheme yields global results in the sense that given any a priori known initial state condition and any output performance requirements regarding the steady state error the speed of convergence and the overshoot the problem is solved. Apart from a sufficient controllability condition for the original system no extra assumptions were made and no extra information was required concerning its nonlinearities. Additionally only the desired trajectory y d t and none of its higher order derivatives was needed. Furthermore the proposed controller is approximationfree and significantly less complex compared with the backstepping approach which has been the main tool for designing controllers for the considered class of nonlinear systems. Finally a comparative simulation study on a third order system verified that the proposed PPC scheme outperforms backstepping in both output performance and required control effort even though in backstepping system nonlinearities were considered known tsec Fig. 3. The required control inputs. The black solid line indicates the control effort of the PPC scheme. The grey solid line indicates the control effort of the backstepping approach. The subplot gives details at the transient. be straightforwardly calculated whereas only the desired trajectory is needed. The selection of the performance function ρ t guarantees a priori that the maximum steady state error will be less than 3 and the speed of convergence of the tracking error will be greater than the exponentialexp 5t.Finallytheselectionofthefunction S ε leads to a nonovershooting response i.e. M =. We simulated both controllers and the results are pictured in Figs The control gains of the backstepping design procedure were selected as k = k 2 = 5 k 3 = such that the response of the tracking error was similar to the exponential e 5t with reasonable control effort. Although asymptotic tracking was achieved see Fig. 2 the appearance of an overshoot was inevitable. Further overshoot reduction can be succeeded by increasing the control gains k i i = 23 leading however to high control input values. On the contrary in PPC the control gains were selected as k = k 2 = k 3 = 2 resulting in a control effort see Fig. 3 whose maximum magnitude appears reduced by half compared to what backstepping requires. Concurrently prescribed performance requirements are achieved see Fig. 2 irrespectively of the fact that the system is considered completely unknown. REFERENCES Bechlioulis C.P. and Rovithakis G.A. 28. Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance. IEEE Transactions on Automatic Control Bechlioulis C.P. and Rovithakis G.A. 29. Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems. Automatica Farrell J.A. and Polycarpou M.M. 26. Adaptive Approximation Based Control: Unifying Neural Fuzzy and Traditional Adaptive Approximation Approaches. Wiley New York NY. GeS.S.HangC.C.LeeT.H.andZhangT.22. Stable Adaptive Neural Network Control. Kluwer Boston MA. Ge S.S. Hong F. and Lee T.H. 24. Adaptive neural control of nonlinear time-delay systems with unknown virtual control coefficients. IEEE Transactions on Systems Man and Cybernetics Part B: Cybernetics Krstic M. Kanellakopoulos I. and Kokotovic P.V Nonlinear and Adaptive Control Design. Wiley New York. Ryan E.P. 99. A universal adaptive stabilizer for a class of nonlinear systems. Systems and Control Letters Spooner J.T. Maggiore M. Ordonez R. and Passino K.M. 22. Stable Adaptive Control and Estimation for Nonlinear Systems-Neural and Fuzzy Approximator Techniques. Wiley New York. 4. CONCLUSIONS 3