Backstepping Control of Linear Time-Varying Systems With Known and Unknown Parameters

Transcription

1 1908 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 11, NOVEMBER 2003 Backstepping Control of Linear Time-Varying Systems With Known and Unknown Parameters Youping Zhang, Member, IEEE, Barış Fidan, Student Member, IEEE, and Petros A Ioannou, Fellow, IEEE Abstract The backstepping control design procedure has been used to develop stabilizing controllers for time invariant plants that are linear or belong to some class of nonlinear systems The use of such a procedure to design stabilizing controllers for plants with time varying parameters has been an open problem In this paper we consider the backstepping design procedure for linear time varying (LTV) plants with known and unknown parameters We first show that a backstepping controller can be designed for an LTV plant by following the same steps as in the linear time-invariant (LTI) case and treating the plant parameters as constants at each time Its stability properties however cannot be established by using the same Lyapunov function and techniques as in the LTI case We then introduce a new parametrization and filter structure that takes into account the plant parameter variations leading to a new backstepping controller The new control design guarantees exponential convergence of the tracking error to zero if the plant parameters are exactly known If the parameters are not precisely known but the time variations of the parameters associated with the system zeros are known, the appropriate choice of certain design parameters, without using any adaptive law, leads to closed-loop stability and perfect regulation This new control design is modified and supplemented with an update law to be applicable to LTV plants with unknown parameters In the adaptive control design, the notion of structured parameter variations is used in order to include possible a priori information about the plant parameter variations With this formulation, only the unstructured plant parameters are estimated and are required to be slowly time varying, and the structured plant parameters are allowed to have any finite speed of variation The adaptive controller is shown to be robust with respect to the unknown but slow parameter variations in the global sense We derive performance bounds which can be used to select certain design parameters for performance improvement The properties of the proposed control scheme are demonstrated using simulation results Index Terms Adaptive control, backstepping, certainty equivalence, parametric robustness, structured parameter variations, time varying systems NOMENCLATURE The following notation is used throughout this paper, unless otherwise stated th element of vector th coordinate column vector in th row of matrix Manuscript received October 25, 2002; revised May 5, 2003 Recommended by Associate Editor P A Iglesias This work was supported by the National Science Foundation under grant ECS Y Zhang is with Synopsys, Inc, Mountain View, CA USA ( youping@synopsyscom) B Fidan and P A Ioannou are with the Department of Electrical Engineering, University of Southern California, Los Angeles, CA USA ( fidan@uscedu; ioannou@uscedu) Digital Object Identifier /TAC Matrix Frobenius norm of Estimate of scalar or vector signal Estimate error shifted truncated -norm( -norm), Set, for a given constant, are some finite constants, and is independent of for a given function, are some finite constants Mean-square-error (MSE) norm of function, Any exponentially decaying to zero signal Any positive constant ( ) identity matrix I INTRODUCTION RESEARCH on adaptive nonlinear control has been accelerated during the last decade, after introduction of a class of controllers for a set of general classes of nonlinear systems [1] [7] These controllers are based on integrator backstepping together with other nonlinear design tools such as nonlinear damping [1], [7], [8], tuning functions [7], [9], and and MT filters [4], [7], [10], [11] In the absence of modeling uncertainties, these controllers achieve global boundedness, asymptotic tracking, passivity of the adaptation loop irrespective of the relative degree, and most importantly, systematic improvement of transient performance [7], [12] These controllers have later on been modified so that they can tolerate a class of modeling uncertainties, especially high frequency unmodeled dynamics, in the global sense [13] [16] The set of systems which can be controlled by these controllers includes linear time-invariant (LTI) systems Moreover, for LTI systems, these controllers bear strong parametric robustness in the sense that global stability can be achieved by choosing appropriate design parameters without the precise knowledge of the values of the plant parameters [6], [7], [17] The corresponding adaptive controllers which deal with unknown but constant parameters [9], [7] can achieve arbitrarily improved transient performance [7], [12] /03$ IEEE

2 ZHANG et al: BACKSTEPPING CONTROL OF LINEAR TIME-VARYING SYSTEMS 1909 The stability properties of this class of controllers are based on the assumption that the plant parameters are time invariant (TI) In most applications, however, plant parameters may vary with time and therefore the properties of the controllers that are designed for LTI plants need to be evaluated in a time varying (TV) environment The early attempts to design adaptive controllers for linear time-varying (LTV) systems are based on the use of the certainty equivalence approach that combines a controller structure with a robust adaptive law [18] [21] These controllers use the notion that slow time variations of the plant parameters act as a perturbation to the system in the same manner as unmodeled dynamics Based on this notion, robust adaptive control schemes for LTI systems are used to guarantee signal boundedness and small tracking error of the order of the time variations of the plant parameters Later, consideration of the TV nature of the plant and some a priori information about the parameter variations led to new adaptive model reference and pole placement control designs that allow the system to be fast TV [22], [23], [21] These controllers bear the strong stability and robustness properties of their traditional counterparts for LTI systems However, they can not guarantee good transient behavior [24], [25], and generally can not be extended to nonlinear time varying systems In this paper, we fill this gap using the backstepping control design procedure We first consider the use of the backstepping controllers proposed in [6], [17] based on TI models to control LTV systems with known parameters by treating the time varying parameters as constant at each time We demonstrate that the quadratic Lyapunov function-based analysis used in [6], [17] to show stability and asymptotic tracking for LTI systems does not work for LTV systems in general, even when the plant parameters are known exactly at each time In addition, we establish that signal boundedness can only be guaranteed if the plant parameters associated with the plant zeros vary slowly with time We, then, propose a new controller that guarantees stability and convergence of the tracking error to zero independent of the speed of variation of the plant parameters The new controller uses integrator backstepping and nonlinear damping and exploits the fact that the TV plant parameters and their variations are known exactly The stability and performance of the proposed controller is examined in the presence of parametric uncertainty The controller guarantees signal boundedness provided that the time variations of the parametric uncertainty associated with the plant zeros are small In particular, if we know the time variations of these parameters exactly, then exponential regulation can be achieved for zero reference input The new controller developed for the known parameter case based on the LTV plant model is modified and combined with an adaptive law to deal with the case of unknown plant parameters The notion of structured parameter variations is used to incorporate any available a priori information about the modes of variation of the plant parameters into the parameter estimates The resulting adaptive backstepping controller has the following advantages as applied to LTV plants First, only the unstructured plant parameters are estimated and are required to be slowly TV The structured plant parameters are allowed to have any finite speed of variation Second, the performance bounds derived can be used to choose certain design parameters for improved performance Furthermore, we show that the proposed adaptive controller is robust with respect to unknown but slow parameter variations Finally, we demonstrate the properties of the developed controllers using simulations II PROBLEM STATEMENT A single-input single-output (SISO) linear plant with parameters that are smooth and bounded and have bounded derivatives which is strongly controllable and observable is topologically equivalent to the following observable canonical form [21], [26], [27]: (21) (22) The state space model (21),(22) can also be represented in the input output form (23) are the right polynomial differential operators (PDOs) [19], [21] for the plant Equivalently, (23) can also be represented using the right polynomial integral operator (PIO) as We make the following assumptions about the plant Assumption 1 The PIO is exponentially stable with a rate at least, cor- satisfies for some constant ie, the transition matrix responding to Assumption 2 The PDO s, are strongly coprime with known orders,, respectively, and Assumption 3 The plant parameters, are time functions which are bounded and have bounded derivatives Assumption 4 The sign of the high frequency gain is known and constant, and there exists a known constant such that

3 1910 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 11, NOVEMBER 2003 The control objective is to design an output feedback control law so that the closed-loop system is uniformly stable, and the plant output tracks as close as possible a bounded, continuously differentiable reference signal with measured bounded derivatives up to order For LTI systems, is a constant vector In our case, however, is a vector function of time The TV nature of does not affect the form of the observer equation given by (32) which is the same as that with constant The observation error equation, however, is given by III BACKSTEPPING CONTROL: POINTWISE DESIGN Let us assume that the plant parameters are known at each time and use the backstepping approach to design a control law that could meet the control objective if the plant parameters were frozen in time at each point in time We refer to this design as pointwise in time In other words, we use the backstepping design approach developed for LTI plants to a plant that is considered for design purposes to be an LTI plant at each frozen time in the parameter space Then we examine whether such a design approach can lead to a controller that can handle the parameter time variations We consider the state dynamics (21) to construct a state estimator Selecting a design vector such that the matrix (or the polynomial ) is Hurwitz, we can rewrite (21) in the Laplace domain (assuming frozen parameters at each time )as It is clear that in the LTI case, is a constant vector, as Since is TV, ie,, can no longer be guaranteed to go to zero in general Let Then, a plant parameterization to be used for control design is obtained by differentiating and using (32) as follows: Introducing the notation Hence, we can use the following equation to derive a state estimator: (31) Assuming that we have no a priori information about the state vector, we can set to zero and expand (31) as Treating the plant as LTI (the plant coefficients as constant), we obtain Denoting, ( ), and ( )by,, and, respectively, we obtain a state estimation scheme which is very similar to [6], [7], and [17] (32) we can write (35) The time variations affect the plant parametric (35) through the signal that also depends on the filtered values of, Since is not considered to be known, it can only be treated as a modeling error The backstepping control design based on (35) with is given as follows: (36) (37) (38) (39) (310) (311) Noticing that for and, we can generate ( ) and ( ) using the following filters:, are positive design constants and (312) (33) (34) The control law is if if (313)

4 ZHANG et al: BACKSTEPPING CONTROL OF LINEAR TIME-VARYING SYSTEMS 1911 Let us analyze the stability properties of the controller (313) designed for when applied to the TV plant with by considering the following Lyapunov function: which has been used in the LTI case Its derivative can be computed using (35) (311) as As we can see, if, then, and will decay to zero exponentially fast, noting that is the generic notation for exponentially decaying to zero signals and for any constant, which can be chosen arbitrarily small However when, due to the presence of which depends on and, signal boundedness is not guaranteed let alone asymptotic tracking unless the time variations are sufficiently small or decay to zero with time The above analysis demonstrates that, in the presence of plant parameter variations, we can not prove stability with the control law based on the backstepping approach for LTI plants using the quadratic Lyapunov function based analysis of [6] and [17] In the following section, we modify the backstepping control design to take into account the time varying nature of the plant parameters IV BACKSTEPPING CONTROL: TIME VARYING DESIGN In this section, we use the backstepping procedure for control design by taking into account the fact that the plant is TV As before, we assume that the plant parameters are known at each time A Observer Design for the Time Varying Plant The reason that the controller (313) can not guarantee perfect tracking or even global stability is due to the term in the parameterization (35) The signal, which acts as a perturbation to the closed-loop system, is due to the time variation of the parameter vector and depends on the closed-loop signals,, and is therefore not guaranteed to be vanishing or even bounded However, can be constructed as follows if is known Consider the filter (41) and define (42) It can be easily verified that with defined in (42) satisfies and, therefore, converges to the true state as If is not known then in (42) can be generated from (43) which follows from (41) by applying the linear swapping lemma [21], [28] Combining (43) with (42), and denoting and by and, respectively, we obtain The signals,, and can be generated using the filters The number of the filters can be reduced by defining, ie, combining (45) and (46) as follows: (44) (45) (46) (47) (48) The filter (47) is used for backstepping design purposes It is easy to verify that the estimation error satisfies (49) which indicates that, and therefore exponentially Using (47), (48), and (44), the following plant parameterization is obtained: (410) Note that the observer (44) incorporates the TV parameters in the filter design which gives us the desired observation error (49) In addition, only two filters are used, hence, this observer scheme has the potential of reducing the mathematical complexity of the control law However, we also note that the number of th order filters required for observer (44) is three, which is one more than that in the LTI case This is for compensating for the time variations of the plant parameters and achieve perfect tracking In the following section, we apply the backstepping procedure to design a controller for (21) and (22) based on observer (44) and parameterization (410) that are more suitable for LTV plants B Backstepping Controller Design Let us apply the backstepping controller design steps to the LTV plant given by (410) Step 1) We treat as the first virtual control We define and choose (411) (412) Step ) In each subsequent step, we individually treat as the virtual control and, therefore, the associated error signals and stabilizing functions are recursively defined as (413)

5 1912 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 11, NOVEMBER 2003 complexity of the stabilizing functions This is another advantage of the new controller The results of Theorem 41 are based on the assumption that the TV plant parameters are known for all In many applications, this assumption may not hold Consequently, it is of interest to examine the robustness of the proposed controller when only some nominal (TV) values of the plant parameters are known In the following section, we address the robustness of the controller with respect to parametric uncertainties (414) In the final step, when we differentiate, the control appears in the form of Therefore, we can design the control law as if if (415) is the th stabilizing function bearing the same definition as (414), which completes the design The stability of the control law (415) can be established by using the Lyapunov function (416) V PARAMETRIC ROBUSTNESS OF THE PROPOSED CONTROLLER In the previous sections, we assumed that the plant parameters were known exactly A natural question one may ask is: What if the plant parameters are not precisely known? That is, what if the actual plant dynamics are described by (51) instead of (21), and there exist errors, between the actual parameters, and the parameters, used in the control design? This section answers this question Due to the parameter errors and, the observer described in (42) or (44) is no longer a true state observer In fact, if we substitute and in (44), we obtain whose derivative is given by Applying the swapping lemma, we have (417) for any constant, which can be chosen arbitrarily small From (416) and (417), it follows that as exponentially fast Hence, the tracking error converges to zero exponentially, and the signals and are uniformly bounded To establish the boundedness of, we first see that is bounded due to the exponential stability of Using the boundedness of, we can recursively establish that and finally are all bounded We summarize our results using the following theorem Theorem 41: For the LTV plant (21) and (22) with Assumptions 1 4, the controller given by (415) guarantees that the closed-loop system is internally stable, and the tracking error converges to zero exponentially fast We note that the traditional polynomial based model reference controller scheme cannot guarantee perfect tracking when the TV plant parameters are completely known [19], and that a different filter structure has been proposed in [22] to resolve this problem for the model reference control case In our case, perfect tracking is achieved using two th order filters We also notice that by using only the signals and instead of a series of s and s, we have significantly reduced the mathematical The corresponding plant parameterization is (52) A lengthy analysis based on the plant parameterization (52) results in the following theorem, which establishes the robustness properties of the controller (415) with respect to the parametric uncertainties Theorem 51: Assume that the parameter error remains small for all time in the sense that such that Furthermore, select the design parameters, to satisfy (53)

6 ZHANG et al: BACKSTEPPING CONTROL OF LINEAR TIME-VARYING SYSTEMS 1913 and (54) (55) VI ADAPTIVE BACKSTEPPING CONTROL In the previous sections, it is assumed that the plant parameters are known precisely or with some small error In this section, we consider these parameters as unknown functions of time which satisfy Assumptions 1 4 In order to incorporate any available a priori information about the modes of variation of the plant parameters, we use the structured parameter variations representation [21], ie, we assume that the plant parameters, have the following known structure: for, is a statespace realization of the equation shown at the bottom of the page is the positive definite symmetric solution to the Lyapunov equation, and is a constant positive definite matrix satisfying Then, there exists a such that, if the derivatives of the parameter errors satisfy for some,, then the closed loop system (51), (22), (415) is uniformly stable, and the tracking error is of the order of in the mean square sense Proof: The proof is long and technical, and is presented in Appendix A Remark 51: Theorem 51 indicates that the uncertainty in the parameters can be counteracted by increasing the values of the design parameters,,,, in particular Note that for sufficiently small, we can find design parameters, such that (53) (55) hold As for the parameters, only the derivatives of the parameter errors have to be small, not necessarily the parameter errors themselves This suggests that the backstepping controller has strong parametric robustness as opposed to the traditional ones A special situation is the LTI case, the time variations of the plant parameters or parameter errors are zero Then, we reach the same conclusion as in [6] In this case, if the reference input is zero, then exponential regulation is achieved In this section, we assume that the nominal values of the TV plant parameters are known For stability, we require that the parametric uncertainty is small in the sense that the time variation (first time derivative) of the parametric uncertainty is small in the average sense, ie, small most of the time In the following section, we combine the proposed controller designed for LTV plants with known parameters with an appropriate parameter estimation scheme to deal with the case of unknown plant parameters (61), form the decomposition of which is a matrix of known time functions, is the unstructured parameter vector that is unknown; is a known parameter vector which can be decomposed to Note that the leading rows of, are zeros Furthermore, we assume the following about the leading nonzero term of and the unstructured parameters Assumption 5: The sign of is the same as the sign of for all Moreover, the unstructured parameter vector is differentiable with respect to time and satisfies, ie, the signals are bounded and and for some and, a small scalar Assumption 5 requires the mean square value of the time variations to be of order, will be required to be small Next, we exploit the TV model based filter design of Section IV to construct a state estimator for the unknown parameter case Using (61), we can rewrite (47) and (48) as Applying the linear swapping lemma, we get

7 1914 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 11, NOVEMBER 2003,,, and Hence, constructing the filters (62) (63) (64) (65), is such that is exponentially stable with stability margin, ie, has all eigenvalues nonpositive, and considering the virtual observer (66) linearly independent time functions, then the filter can be obtained using th order filters For example, suppose, s are constant matrices, are linearly independent scalar time functions, then it suffices to implement The matrix, can be realized using as it is straightforward to verify that the observation error satisfies When, as Hence, (66) is a true state observer for (21), (22) when the parameter vector is known and constant If is not constant, then the observation error is nonvanishing and is represented in the following transfer function form: (67) Using (67), we can obtain the following plant parameterization: Moreover, if, can be linearly represented by,, respectively, say, are constant vectors of length, then the filter signals and can be obtained through, as follows: (68) Similar to the pointwise design of Section III, the parameterization (68) appears to be in the same form as the LTI case [7], [17] except for the term in, and is suitable for applying the adaptive backstepping design Remark 61: Note that (61) covers the general case including the fully structured, unstructured, and known parameter cases If the parameters are unstructured, then we simply have If the parameters are fully structured, then is constant but unknown The case corresponds to the known parameter case Remark 62: Even though the filters (62) and (63) appear to be of high order since both, are matrices, the actual implementation of these two filters can be of lower order, depending on the elements of, In general, if contains In this case the filters are of total order In particular, when the parameters are not structured, and, then the total filter order is, which is similar to the LTI case [6], [7], [17] A Certainty Equivalence Control Law The controller design follows the same procedure as in the known parameter case The idea is to recursively treat as a virtual control signal, and apply the backstepping procedure using the certainty equivalence, ie, replacing the unknown parameter vector with its on-line estimate The design steps are as follows Step 1) Step ) (69) (610) (611)

8 ZHANG et al: BACKSTEPPING CONTROL OF LINEAR TIME-VARYING SYSTEMS 1915 suitable for synthesizing an adaptive law based on a Lyapunov function We define the auxiliary filter and the auxiliary error signal (616) Then, the error signal satisfies the equation (612) Since is not guaranteed to be bounded, we introduce the following normalizing signal: (613) In Step ), the control appears in the form of, therefore the control law can be chosen as if and are design constants Some important properties of are given by the following lemma Lemma 61: We have (617) (618) if (614) is either 0 or 1, the latter corresponding to the case appears explicitly in the control law Note that for the control law (614) to exist, the adaptive law must assure that With the control law (614), the corresponding error system is given by is a positive constant Proof: The state (21) can be rewritten as from which we obtain (619) (620), (615) Observing (62) (65) and (620), we see that,,,, and can be represented as outputs of stable filters with inputs and Hence, the result (617) follows immediately Similarly, (67) implies that, which together with (617) leads to (618) Finally, using the inequality, we obtain Now, define the normalized estimation error Then, satisfies B Adaptive Law With an Auxiliary Filter The adaptive law for generating the parameter estimates used in the control law (614) is based on the idea of introducing an auxiliary filter to counteract the effect of term in the equation, therefore ending up with a new error system that is (621) By considering (621), (619), and the following Lyapunov-like function: (622)

9 1916 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 11, NOVEMBER 2003 the following robust adaptive law can be chosen: we obtain (623) is the nominal value of, ie, is expected to be close to the constant vector, is a positive definite symmetric gain matrix, is the projection operator to project along the boundary, which is defined as (626) In view of (618), if, hence, (626) implies that In addition, integrating (625), we get that and if and otherwise is the leakage coefficient [28] defined as if if (624) is a known upper bound for, and is a small constant The stability properties of the adaptive law are described by the following lemma Lemma 62: Assume that, then the adaptive law (623) guarantees the following i),, ii) iii),,,,, In particular, if, then and as Proof: The properties in i) are direct consequences of the projection and switching -modification, see [28] To prove ii) and iii), let us consider the Lyapunov-like function (622) For simplicity and without loss of generality, we assume The derivative of along the solution of (621), (623) can be computed as Using and, it follows that, and, consequently, using i) and ii), it follows that Due to the linearity of the stabilizing functions, depends only on the parameter estimates and is linear in,,,,, thus, and follows from (623) and Using (616), satisfies (627) from which follows immediately If, then and all the properties become properties, ie, Finally, using (621) and (627) we see that, which together with implies that as Having established the stability properties of the adaptive law, we analyze the closed-loop stability properties of the adaptive control scheme based on the error system (615) next The following theorem summarizes the results of this analysis, which is presented in Appendix B in details Theorem 61: The adaptive controller described by (614) and (623), when applied to the LTV plant (21),(22), guarantees the existence of a constant such that, all the closed-loop signals are uniformly bounded, and the tracking error is of the order in the mean square sense, ie, (628) (625) and are finite positive constants Moreover, can be expressed as Using the inequality and is a finite positive constant independent of,, Proof: The proof is similar to that of Theorem 51 and is presented in Appendix B Theorem 61 indicates that, using the adaptive controller (614) and (623), only the time variations of the unstructured plant parameters are required to be small in the mean square sense to guarantee closed-loop stability and tracking with small

10 ZHANG et al: BACKSTEPPING CONTROL OF LINEAR TIME-VARYING SYSTEMS 1917 MSE The overall system is not necessarily restricted to be slowly TV The requirement about the time variations of the unstructured plant parameters is necessary since it is not possible to estimate unknown and arbitrarily fast TV parameters using a general adaptive law with finite speed of adaptation [22], [28] Once this requirement is satisfied, the mean square tracking error is guaranteed to be of the order of the speed of the unstructured plant parameter variations Besides establishing stability and tracking properties, the theorem provides guidelines for performance improvement as well It shows that the MSE performance can be improved by amplifying,, and possibly for small enough to satisfy the stability conditions Unlike [12], arbitrary performance improvement is only assured in terms of the normalized tracking error The bound on depends on However, although might increase by increasing,,, this can be counteracted by reducing the normalization coefficient Hence, the performance of the adaptive backstepping controller can be improved by adjusting the design parameters,,, and We demonstrate this fact via simulations in Section VII C Fully Structured Parameter Variations The case of fully structured parameter variations corresponds to being constant We generalize it to the situation that For this special class of LTV plant, the proposed adaptive controller (623) and (614) has the following properties Corollary 61: If the speed of parameter variations satisfy, then the adaptive controller (614) and (623) guarantees that all the closed-loop signals are uniformly bounded, and the tracking error converges to zero asymptotically Proof: This is a direct consequence of Theorem 61 and Lemma 62 iii) Due to the transformation (61), the parameter vector may not reflect the plant parameters themselves, and can contain more or less than elements, which corresponds to the overparameterized and the underparameterized case, respectively Corollary 61 indicates that when full knowledge of the parameter variations is available, then regardless of the speed of the parameter variations of the plant, global stability is guaranteed, and asymptotic tracking is achieved In addition, in the case of fully structured parameter variations, is exponentially vanishing Therefore, in this case, we can apply the tuning design given in [6], [17] instead of the certainty equivalence approach using parameterization (68) The advantage is a guaranteed performance improvement, as in the TI case [7], [12] VII SIMULATION RESULTS Let us consider a simple unstable second-order LTV plant whose state-space representation is (71) (72) Fig 1 Response using the pointwise design and exact knowledge of g, g, g (a) c = d = c = d =1 (b) c = c =5, d = d =1 (a) (b),, and It is required to design a controller so that the output tracks the reference signal Let us first apply the pointwise design, ie, the control scheme (36) (313) together with the estimation filters (33) and (34), assuming that,, are all known exactly Noting that,, for the plant, the filter parameters are chosen as, and the design parameters are chosen as Fig 1(a) shows the result Although the output signal is bounded, tracking performance is not that successful Next, we increase the values of the design constants and to 5 Tracking performance is enhanced as shown in Fig 1(b) However, asymptotic tracking is not achieved Then, we repeat the same simulations with the LTV design ie, the control scheme (411) (415) together with the estimation filters (45) (47) Tracking is perfect as shown in Fig 2 Next, we consider some parametric uncertainty We assume that our plant model is a little bit erroneous, eg, models of the actual functions,, of the plant are, Choosing the design parameters as,, we can see from Fig 3 that the system is stabilized, and a relatively small tracking error (smaller than that of controller (313) with known parameters) is obtained

11 1918 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 11, NOVEMBER 2003 (a) (a) (b) Fig 2 Response using the LTV design and exact knowledge of g, g, g (a) c = d = c = d =1 (b) c = c =5, d = d =1 (b) Fig 4 Adaptive control with different choices for c, c, 0 (d = d =1, =2, =1) (a) Tracking (b) Parameter estimation Finally, we consider the unknown parameter case assuming that the plant structure (71) (72) is known but the functions,, are unknown In order to build up an adaptive controller, we first write the plant parameter in structured parameter variations form as follows: Fig 3 Response using the LTV design in presence of parametric uncertainty (g (t) =1, g (t) = g (t) =2, c = c =5, d = d =1) Note that the parametric uncertainty has amplitude 1, however, its derivative has a much smaller amplitude 01 This demonstrates that the time variation of the uncertain parameter, not the size of uncertainty itself, determines the system stability and performance Noting that and hence are zero, the estimation filters are implemented using (62), (64), and (65) The following

12 ZHANG et al: BACKSTEPPING CONTROL OF LINEAR TIME-VARYING SYSTEMS 1919 (a),,,,,, are design constants, is the switching- coefficient defined in (624) Fig 4 shows the tracking error and the parameter estimate for simulations with different choices of the design parameters and the adaptive gain parameters The switching and normalization parameters are set as,,, in all of these simulations The response for,, is redrawn in Fig 5(a) in order to make it comparable with the results for the cases with known and unknown parameters As can be seen in these figures, the system is stabilized, and the tracking error remains in a neighborhood of 0, for all design parameter choices As can be seen in Fig 4, increasing the value of the design parameters improves the tracking performance as in the known parameter case By increasing the adaptive gain, not only parameter estimation gets faster but the tracking performance is improved further as well Later, fixing,,,,,, the effect of the normalization coefficient is tested The results in terms of the tracking errors are shown in Fig 5(b) As seen in this figure decreasing has a similar effect with increasing on enhancement of tracking In the aforementioned simulations, we see that the parameter estimates adapt to the parameter changes We have also observed that the control effort remains within a reasonable bound Since the only unknown TV parameter is slowly time varying, stability is guaranteed (b) Fig 5 (a) Response using the adaptive controller (c = c =5, d = d = 1, 0=10, =2, =1) (b) Adaptive tracking with different choices for (c = c =5, d = d =1, 0=10, =2) control law is designed based on the steps in Section VI selecting : The adaptive law and the associated auxiliary signal are defined as VIII CONCLUSION In this paper, we introduced a new backstepping controller for LTV systems with known and unknown parameters The controller guarantees exponential tracking when the plant parameters are known exactly When the plant parameters are not known exactly but their time variations are small enough, regardless of the size of the parameter errors (except for the high frequency gain), global stability can be guaranteed by choosing certain design parameters properly Hence, the proposed controller has strong parametric robustness properties which most of the traditional model reference controllers do not have When the plant parameters are unknown, the proposed controller is combined with an online parameter estimator to form a new adaptive controller This new adaptive controller guarantees the following All the closed-loop signals are globally uniformly bounded The tracking error remains small and of the order of the speed of the unstructured plant parameter variations, which is required to be small in the mean square sense If the plant parameter variations are fully structured, the tracking error converges asymptotically to zero The performance bounds for the tracking error developed can be used to select certain design parameters for performance improvement The expected performance of the proposed controller and the effects of design parameter selections on the transient performance are illustrated by simulation results The proposed controller is suitable for use in many applications the plant parameters are time varying An example of such application is the control of aerospace systems the parameters of the system vary with time and/or flight conditions

13 1920 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 11, NOVEMBER 2003 [29] [31] Application of the proposed backstepping scheme to flight control of high performance aircrafts and hypersonic air breathing vehicles is currently under investigation APPENDIX I PROOF OF THEOREM 51 Using (52), (411), and (414), we obtain the following error equation: (A1) Suppose that is a state-space realization of, then (A3) (A4) Note that and are independent of the design parameters Since is an exponentially stable matrix, there exists a positive-definite matrix such that Define error equation of, then we substitute (A4) into the and get Next, we consider the term state-space form: (A5), which we write in the following Augmenting (A6) with (33), we get (A6) We first consider the term We have (A7) Finally, we augment the error systems (A1) and (A3) with (A7) using (A5) and get (A8) (A2) Let the th order monic polynomial and the th order monic polynomial be a decomposition of, ie, Then, the operator in of (A2) can be written as which is the sum of two proper and exponentially stable I/O operators Hence, the operator in of (A2), which we denote as, is a proper and exponentially stable I/O operator Using (A2), (411), and the definition of, we can write

14 ZHANG et al: BACKSTEPPING CONTROL OF LINEAR TIME-VARYING SYSTEMS 1921 We first analyze the homogeneous part of (A8) by individually considering the following two partial Lyapunov functions: we get are constants to be chosen and is a constant matrix satisfying The derivatives of, along the solution of (A8) are computed as (A9) Let us choose (A10) Using the inequalities is an arbitrary constant, and consider the Lyapunov function is another constant to be chosen Using (A9) and (A10), we have the following: We first pick, are arbitrary constants With these choices, if are arbitrary constants, then Since,,,, are arbitrary, the existence of,, is guaranteed provided that (53) (55) are satisfied Hence, if,, and for

15 1922 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 11, NOVEMBER 2003 are chosen to satisfy (53) (55), then the homogeneous part of (A8) is exponentially stable Now, let us suppose,, and for are chosen as such, and go back to (A8) Since, the corresponding output is also bounded Therefore it suffices to consider the subsystem,, are all zero Next, we define a fictitious normalizing signal (A11) is a constant such that, are exponentially stable The normalizing property of can be described by the following lemma Lemma 81: Regardless of the boundedness of any closedloop signal, we have,, are chosen as before Using (A13) and continuity of Using (A8), (A16), and the fact that (A16) for an arbitrary function, the derivative of can be computed as (A12) for all and some Proof: Using the inequality which follows directly from the definition of (A11), we have that for all for which is exponentially stable (A13) given by Therefore,, all the normalized signals are bounded and small in the order of in the mean square sense In particular, On the other hand, using (36) (312) and (411) (414), we can represent the control law (313) or (415) in the form for some bounded functions,,, Substituting (A17) in (A17), we get (A14) is the impulse response of and Note that is finite since is exponentially stable Using Holder s inequality, we have that and, is a Hurwitz polynomial of order,, Hence, applying [30, Th (22), p 113], we obtain The result follows directly from (A14) and (A15) Now, define the following normalized errors: and consider the Lyapunov function (A15) and are impulse responses of and, respectively Since and are both exponentially stable and,,, are bounded smooth functions of which is a bounded function of time, the supremum terms mentioned before are all finite Hence, using, we get

16 ZHANG et al: BACKSTEPPING CONTROL OF LINEAR TIME-VARYING SYSTEMS 1923 or is a constant such that is an exponentially stable operator Applying the arguments we have used in the proof of Theorem 51 again, we derive that is some constant Applying the Bellman Gronwall Lemma, we have is some finite positive constant Since Now, if and, then and Once is bounded, is bounded Then, we apply the same argument as before, and conclude that all the closed-loop signals are uniformly bounded, and the closed-loop system is internally stable In addition, all the error signals satisfy That is, all the error signals, including the tracking error are of the order of in the mean square sense, ie, we have, ie, such that Applying the Bellman Gronwall lemma, we obtain for any and some constants independent of APPENDIX II PROOF OF THEOREM 61 Following equations (69) (613), it can be easily shown that the control law (614) can be represented as (B1) Let, then,wehave Since bounds which bounds all the closed-loop signals, it follows that all signals are uniformly bounded In addition, the tracking error satisfies In order to get some quantitative results, we first derive some performance bounds for the estimation error We start by calculating the bound of the Lyapunov function Noting that is assumed to be zero for simplicity and without loss of generality, (626) yields for some continuous functions,,, On the other hand, from (23), we get Substituting (B2) into (B1), we obtain (B2) (B3) From (B3), we notice that is a function which decreases with increasing,, Therefore, the bound on, is decreasing with the increase of,, Using the fact that, and,we integrate (625) and get so that is some Hurwitz polynomial of order,, and Let us define a fictitious normalizing signal (B4) (B5) is some positive constant independent of

17 1924 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 11, NOVEMBER 2003 Next, we derive some performance bounds for the auxiliary signal Considering the quadratic function it follows directly from (627) and the definition of Therefore that Since,,, depend only on,,,, and is nonincreasing with the increase of, using (623) and (B5), we obtain (B6) is some positive constant independent of,, Finally, combining (B4) and (B6) we obtain (628) REFERENCES [1] I Kanellakopoulos, P V Kokotović, and A S Morse, Systematic design of adaptive controllers for feedback linearizable systems, IEEE Trans Automat Contr, vol 36, pp , 1991 [2] R Marino, I Kanellakopoulos, and P V Kokotović, Adaptive tracking for feedback linearizable SISO systems, in Proc 28th Conf Decision Control, Dec 1989, pp [3] R Marino, P Tomei, I Kanellakopoulos, and P V Kokotović, Adaptive tracking for a class of feedback linearizable systems, IEEE Trans Automat Contr, vol 39, pp , June 1994 [4] R Marino and P Tomei, Global 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1975 Youping Zhang (M 02) received the BS degree from the University of Science and Technology of China, Hefei, Anhui, ROC in 1992, and the MS and PhD degrees in electrical engineering from the University of Southern California, Los Angeles, CA in 1994 and 1996, respectively From 1996 to 1999, he was a Research Engineer with the United Technologies Research Center, East Hartford, CT He joined Numerical Technologies, Inc, San Jose, CA, in July 1999 and stayed for nearly four years until it was acquired by Synopsys in February 2003 At Numerical Technologies, he was in several different positions including software product development, technology research, and technical marketing for resolution enhancement technologies He is currently a Technical Marketing Manager for Mask Synthesis at Synopsys His research interests are in the areas of optimizations, numerical methods, computer aided designs, and intelligent systems

18 ZHANG et al: BACKSTEPPING CONTROL OF LINEAR TIME-VARYING SYSTEMS 1925 Barış Fidan (S 02) received the BS degrees in electrical engineering and mathematics from Middle East Technical University, Ankara, Turkey in 1996, and the MS degree in electrical engineering from Bilkent University, Ankara, Turkey in 1998 He is currently working on the PhD degree in Electrical Engineering-Systems at the University of Southern California, Los Angeles His research interests include adaptive and nonlinear control, switching and hybrid systems, robotics, high performance and hypersonic flight control, semiconductor manufacturing process control, and disk-drive servo systems Petros A Ioannou (S 80 M 83 SM 89 F 94) received the BSc degree (first class honors) from University College, London, UK, and the MS and PhD degrees from the University of Illinois, Urbana, in 1978, 1980, and 1982, respectively In 1982, he joined the Department of Electrical Engineering-Systems, University of Southern California, Los Angeles, California, he is currently a Professor and the Director of the Center of Advanced Transportation Technologies His research interests are in the areas of adaptive control, neural networks, nonlinear systems, vehicle dynamics and control, intelligent transportation systems, and marine transportation He was a Visiting Professor at the University of Newcastle, NSW, Australia and the Australian National University, Canberra, in fall 1988, the Technical University of Crete in summer 1992 and fall 2001, and served as the Dean of the School of Pure and Applied Science at the University of Cyprus in 1995 He is the author/coauthor of five books and over 150 research papers in the area of controls, neural networks, nonlinear dynamical systems, and intelligent transportation systems Dr Ioannou was a recipient of the Outstanding Transactions Paper Award in 1984, and the recipient of a 1985 Presidential Young Investigator Award