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Robust disturbance rejection controller for systems with uncertain parameters

2019; Institution of Engineering and Technology; Volume: 13; Issue: 13 Linguagem: Inglês

10.1049/iet-cta.2018.5291

1751-8652

Alessandro R. L. Zachi, Carlos Alberto Correia, Jair L. Azevedo Filho, Josiel Gouvêa,

Advanced Control Systems Optimization

IET Control Theory & ApplicationsVolume 13, Issue 13 p. 1995-2007 Research ArticleFree Access Robust disturbance rejection controller for systems with uncertain parameters Alessandro R.L. Zachi, Corresponding Author Alessandro R.L. Zachi alessandro.zachi@cefet-rj.br Graduate Program in Electrical Engineering, Federal Center of Technological Education of Rio de Janeiro, Rio de Janeiro, BrazilSearch for more papers by this authorCarlos Alberto M. Correia, Carlos Alberto M. Correia Graduate Program in Electrical Engineering, Federal Center of Technological Education of Rio de Janeiro, Rio de Janeiro, BrazilSearch for more papers by this authorJair Luiz A. Filho, Jair Luiz A. Filho Department of Control and Automation Engineering, Federal Center of Technological Education of Rio de Janeiro, Nova Iguaçu, BrazilSearch for more papers by this authorJosiel A. Gouvêa, Josiel A. Gouvêa Department of Control and Automation Engineering, Federal Center of Technological Education of Rio de Janeiro, Nova Iguaçu, BrazilSearch for more papers by this author Alessandro R.L. Zachi, Corresponding Author Alessandro R.L. Zachi alessandro.zachi@cefet-rj.br Graduate Program in Electrical Engineering, Federal Center of Technological Education of Rio de Janeiro, Rio de Janeiro, BrazilSearch for more papers by this authorCarlos Alberto M. Correia, Carlos Alberto M. Correia Graduate Program in Electrical Engineering, Federal Center of Technological Education of Rio de Janeiro, Rio de Janeiro, BrazilSearch for more papers by this authorJair Luiz A. Filho, Jair Luiz A. Filho Department of Control and Automation Engineering, Federal Center of Technological Education of Rio de Janeiro, Nova Iguaçu, BrazilSearch for more papers by this authorJosiel A. Gouvêa, Josiel A. Gouvêa Department of Control and Automation Engineering, Federal Center of Technological Education of Rio de Janeiro, Nova Iguaçu, BrazilSearch for more papers by this author First published: 01 July 2019 https://doi.org/10.1049/iet-cta.2018.5291Citations: 4AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This work proposes a mathematical solution for plants with full set of uncertain parameters which is based on the active disturbance rejection control method. This method has been studied for three decades and its main characteristic is the use of an extended state observer to estimate the non-measurable signals of the plant. However, the application of the basic methodology requires the exact knowledge of the system control gain, which is difficult to measure in the case of parametric uncertainties. Recently, some advances have been discussed in the literature. Some results have revealed the capability of them of rejecting the unknown input disturbance and of ensuring the stability of closed-loop transfer functions, even in the case of unknown order and/or relative degrees. However, such proposals have not been effective for large level of parametric uncertainties. The idea of the present work is to relax the requirement of exact knowledge of plant parameters by proposing modifications in the control structure. Mathematical analysis show that the proposed modification does not affect the original purpose of output tracking. The performance of the proposed strategy is compared with other schemes by using numerical simulations. In addition, experimental results on a real cart-pendulum system are presented. 1 Introduction The control of uncertain dynamical systems has received significant attention of the control community motivated by the difficulty of obtaining the exact model for some systems, particularly involving situations in which their parameters are not perfectly known. The problem of dealing with parametric uncertainties is of great interest in the control systems community and represents a challenging benchmark for industrial and academic applications. Among several control strategies proposed in the literature, many contributions have been reported concerning at linear and non-linear plants, such as the recent ones involving backstepping [1], model reference adaptive control (MRAC) [2], and the references therein. In addition to system uncertainties, another fundamental problem in control system designs is the ability of efficient rejection of external disturbances. In past decades, there have been many interesting and relevant control developments contributing for mitigating system disturbances of any kind [3–6]. In particular, in [3] the problem of adaptive sliding mode control (SMC) for a class of interval type-2 fuzzy system is investigated. In [6], it is proposed that an SMC that eliminates the reaching mode and the impacts of unwanted disturbances while tracking a desired reference trajectory. In order to develop linear control laws, strategies which use disturbance estimation have emerged in literature, such as disturbance observer-based control [7, 8] and active disturbance rejection control (ADRC) [9–12]. The idea proposed by the basic ADRC method is to generate estimates of the non-measurable signals of the plant by using an extended state observer (ESO) [13] and use those estimates to generate the control law for the system [10, 14, 15]. The main characteristics of the ADRC method are simpler implementation and good robustness properties against external disturbance, unmodelled dynamics, and parametric uncertainties. Due to the cited advantages, extensions of the ADRC method were developed which allowed its application in other classes of systems. For instance, in [16], the control of multiple input multiple output systems is addressed in which the solution was implemented using dynamic decoupling based on ADRC strategy. The method was tested on an industrial chemical process for demonstrating the performance of the proposed technique. For higher order systems, the technique discussed in [17] proposes the use of reduced order observers in cascade. In such a configuration, the control and estimation parameters are kept the same for each observer. Then, for determining the parameter values, the particle swarm optimisation procedure is used according to an established cost function. In [18], an extension of the ADRC technique is proposed for dealing with non-minimum phase systems. Such extension uses the ADRC basic strategy associated with the model-assisted feedforward control for ensuring a null steady-state error behaviour with a minimum settling time. As can be seen from the recent works involving ADRC paradigm [11, 12, 17, 19–22], plant uncertainties are taken into account in the control design, with the exception of the control gain, which is considered known. However, in many practical applications, such as load transport by mobile robots and/or drones, the value of the load mass may vary during task execution. In such a situation, uncertainty in the control gain may arise because the mass and inertia parameters of the system mathematical model depend on the value of the load mass [23, 24]. Since this sort of uncertainty affects the amplitude of the control signal and/or its direction, more careful attention should be focused on this issue. In this paper, the problem of systems with a full set of uncertain parameters is addressed. An extension of the basic ADRC method is proposed for controlling systems with uncertain control gain. The central idea of the proposed strategy is to introduce a constant gain block in series with the plant output and also a dynamical block in parallel with them, in order to generate a modified input/output plant with known control gain. The main characteristics of the proposed method, in closed loop are: (i) the maintenance of robustness properties with respect to plant parametric uncertainties, (ii) the maintenance of external disturbance rejection capability, (iii) the introduction of a reduced requirement on the knowledge of the plant constant parameters, particularly with respect to the value of its control gain, (iv) the establishment of stability proofs based on the classical control theory for linear systems, and (v) easy implementation due to the use of linear components, such as filters and state estimators. Experimental results are presented to illustrate the method efficiency when applied in the stabilisation problem of an inverted pendulum on a cart system. Moreover, a cascaded control scheme is also adopted in the experimental tests to deal with the non-minimum phase behaviour of the cited mechanism. In summary, the contributions of the present work are: (i) the paper presents an extension of the Standard ADRC strategy, which is denoted as Modified ADRC, that is capable of dealing with systems with a full set of uncertain parameters and with external disturbance rejection; (ii) a comparison of the proposed scheme with other methods are shown and discussed via simulation results; (iii) the mathematical developments and stability proofs are shown in details; (iv) experimental results obtained after applying the Modified ADRC in a real system are also presented and discussed. This paper is organised as follows: the basic ADRC strategy proposed in [25] is reviewed in Section 2, in which its main robustness properties and disturbance rejection capability are discussed. The proposed extension to the basic ADRC method, which we denote as Modified ADRC, is presented in Section 3. A detailed description of the proposed scheme is addressed for a general class of n th order systems with linear and/or non-linear dynamical equations. The stability and convergence properties are demonstrated in Section 4. This is followed by the discussion of some simulations results shown in Section 5, in which the performance is compared with other control schemes, such as MRAC. The experimental results are presented in Section 6 to show the efficiency of the proposed controller in a real stabilisation problem of an inverted pendulum on a cart mechanism. The work conclusion and other discussions are presented in in Section 7. 2 The ADRC basic method – a brief overview In this section, an overview of the ADRC basic method is presented. At first, the control problem is formulated for a general class of dynamical systems, aiming to highlight the applicability of the technique. The idea of adopting such presentation methodology in the upcoming developments is to focus attention in the general design procedures for better understanding of the resulting closed-loop properties. 2.1 Problem statement As can be seen from the generalisation formalisms adopted in several works, such as [11, 20–22, 26, 27], the ADRC paradigm provides a control design methodology that is suitable for large classes of linear and non-linear systems. In this paper, we consider the control design for a general class of n th order dynamical systems (plants) that can be represented in the following form: (1)in which is the output variable, is the input variable, is an external disturbance, is the system state vector, is a constant, that will be denoted by the input control gain, and represents non-linear function of the system. In this work, we use the notation to represent the n th order time derivative of . The function in (1) is usually known in the literature as the plant total disturbance term [10, 11]. In order to simplify notation, we will henceforth represent the function by . The control objective is to force the plant output to track a desired and bounded trajectory , at least asymptotically, by designing a stable control law for . From input/output point of view, the plant represented by (1) can be considered as a n th order integrator system with an input , an output and an input disturbance . 2.2 Control design By defining the output error as (2)and by assuming the availability of all plant signals and coefficients, a state-feedback control law that would achieve the desired output tracking, could be chosen as in which, the real constants are the coefficients of a n th order monic and stable polynomial given by (3) In such ideal situation, the resulting closed-loop dynamics is exponentially stable and governed by (4) However, in general applications, the total disturbance term (1) may involve some uncertain coefficients and/or may represent a combination of external disturbances, non-measurable signals, and/or unmodelled dynamics of the plant. In such cases, the control law (3) cannot be directly computed. In order to overcome this drawback, an ESO is designed to estimate the function . The basic procedure is to define the plant state vector from (1) including as an additional state variable (5) By assuming that is differentiable, the plant state–space representation assumes the following form: (6) Remark 1.The first notable property of the state–space representation in (6) is that the pair (A,C) is always observable, which is fundamental to the ESO design. Such structural property is intrinsically linked with the plant general description in (1) and occurs independent of the system order n. By assuming, initially, a known control gain b, it is possible to design a full-order ESO as follows: (7)where represents the estimated state vector, is defined as the output estimation error, and is the vector of the observer adjustable gains that are usually chosen as the coefficients of the stable polynomial given by (8)in which is defined as the observer characteristic roots of multiplicity . Then, a realisable version of the state feedback control law (3), according to (5)–(7), can be chosen as (9) 2.3 Stability and convergence analysis Defining the ESO state estimation error as (10)then, from (6) and (7), its closed-loop dynamics yields (11) The output error dynamics, which is obtained after replacing the control law (9) in the system (1), is governed by (12)which was obtained after manipulating from (10). For analysis purpose, let us define the output tracking error vector as (13) By adopting the controllable canonical form for writing the state space representation, the closed-loop full error system, composed by (11) and (12), is described by (14) (15)with (16) In [28, 29], Zheng et al. have established convergence proofs for the ESO error system (14). In fact, Theorem 1 stated in [28] demonstrates that, for a bounded and a finite time instant , the ESO steady-state errors tend to residual sets around the origin, with constant upper bounds given by (17) (18)in which l is some positive integer. A complete proof for (17) and (18) can be found in [28, 29]. In (15), the transfer function from to can be computed by using (19)which results in (20) Note that the plant output error dynamics (15) has a behaviour analogous to the dynamics of a linear filter from to . Then, the norm of the error output will be given by (21) Notice from (3) that . Thus, for a finite time instant , the steady-state behaviour of can be described by (22) Remark 2.It was shown that the estimation errors of ESO in (14) are bounded and that the corresponding upper bounds can be made monotonously small, by increasing the ESO bandwidth. This is achieved by choosing sufficiently large values for the modulus of the design constant (8). Remark 3.Here, it is important to mention the existence of other control schemes based on state feedback with observers. Probably, the most famous are the high-gain observer (HGO)-based ones. The basic difference between them and the present ADRC is whether the uncertainties and/or generalised disturbances of the system, represented here by , are estimated or not. In ADRC, the ESO is designed to estimate and cancel these unwanted terms. Unlike ESO, in HGO-based strategies, the uncertainties and/or generalised disturbances are not estimated. In fact, as shown in [30], HGOs adapted for overcoming these disturbances in the closed loop. Remark 4.Note from (7) and (9) that the ESO design and the control law definition are both dependent on the exact knowledge of the plant control gain b. Such design feature has been addressed in several ADRC schemes, by assuming that b is completely or partially known a priori [9, 10, 12, 22, 31–33]. For plants with uncertain b, few ADRC-based schemes have been reported. This drawback is addressed in the present work, in which a modified design method is proposed that does not require the exact knowledge of b. As will be shown by the following developments, only the sign of the control gain b will be required. 3 The proposed ADRC with modified framework Consider the class of dynamical systems defined by (1). For analysis purpose, let us detach the system linear part from so that it can appear explicitly in the plant representation, as follows (23) (24) In (23), denotes the system constant parameters, whose elements can be either positives, negatives, or null. Before proceeding with the design procedures for the tracking control problem originally stated in Section 2.1, some assumptions are considered: Assumption 1.The disturbance signal and the non-linear function are both bounded and have uniformly bounded first order derivatives : (25)in which are known positive real constants. Assumption 2.The plant parameters are uncertain, but upper bounded by a known constant : (26) Assumption 3.The control gain b is uncertain but has known sign and a known lower bounded constant given by (27) Assumption 4.The reference trajectory and its higher order derivatives are uniformly bounded functions . The Assumption 1 assumes that the amplitudes of the signals and , and also their rates of change, are bounded. Such a priori knowledge is fundamental for the choice of the state estimator bandwidth. The case in which they are allowed to change very rapidly requires the observer bandwidth to be chosen largely for achieving accurate estimates of them. As also concluded in [28], the lack of boundedness assumption on and on could allow their rate of change to be unbounded, which would be very difficult to estimate. By assuming Assumptions 2 and 3, the present work aims to consider a more general class of plants that possess a full set of uncertain parameters. In fact, by considering such reduced knowledge about the system is an important step toward performing less conservative statements in the design of the proposed controller. Assumption 4 is considered for simplicity. 3.1 Proposed methodology The main idea proposed in this work is to perform a structural transformation on the original system (23), in particular concerning the input/output behaviour, in order to obtain a new dynamical system with an advantageous format. For this end, we introduce an adjustable constant gain in series with the plant output error, and a linear stable filter in parallel with them, as shown in Fig. 1. Fig. 1Open in figure viewerPowerPoint Block diagram of the proposed solution In the present proposal, the positive design constant is chosen such that results in a stable polynomial. Based on the configuration of Fig. 1, the new output error can be written as (28) (29) (30) (31) (32) By differentiating (28) n times, the dynamics of the new output error variable , now with , will be given by (33) (34) By highlighting from (28), we have that (35) Then, by replacing (35) into (34), and also using (13) and (31), we obtain (36) (37) For writing the new plant description using the ADRC formalism, as done in (1), we define a new generalised disturbance function as (38)which reduces (34) to (39) By comparing (39) to (1), it is possible to verify that the original problem of output tracking stated in Section 2.1, which is associated with the error (29), is now redefined in terms of the new output error . It is important to notice that the new control input has a unitary coefficient, meaning that the controller and ESO designs can be carried out without requiring the exact value of the original parameter b (23). Since is not available, the basic ADRC design procedures can be addressed in the next section. 3.2 ESO design For (39), consider the following state variable definitions (40) By assuming that is differentiable, the plant state–space representation of (39), in companion form, can be written as (41)with (42) The pair in (41) and (42) which results from the adoption of the modified ADRC formalism, has the same observability property highlighted by Remark 1. Thus, since the pair is always observable, a full-order ESO for (41) and (42) is then designed as follows: (43)where represents the estimated state vector, is defined as the output estimation error and is the vector of the observer gains defined by (44) By defining the ESO state error as (45)the closed-loop dynamics can be computed from (41)–(43), resulting in (46) 3.3 Control design By observing the new plant description in (39), we note that its homogeneous part (i.e. with ) is stable. Therefore, in order to compensate for the disturbance term , forcing the new error to tend to zero, we propose the following control law: (47) In the next section, the stability and convergence properties are discussed for the closed-loop system obtained with the modified ADRC framework. 4 Stability analysis By replacing the control law expression of (47) with (39), the closed-loop dynamics for the error becomes (48) Since the left-hand side of (48) corresponds to a linear, time-invariant, and stable ODE, a bounded will result in a bounded . Provided that the boundedness and convergence properties of the closed-loop signals in (48) are dependent on the ESO estimation error , an investigation about it is needed. In order to verify the influence of the generalised disturbance term in the amplitude of the observer estimation error in (46), let us compute the transfer function from to by using (49)with . For a th order ESO, the computation of (49) can be easily accomplished by using math softwares with symbolic resources. However, due to the particular matrix structure of (46), such a task becomes quite simplified. For instance, considering the case of plants of order , it is not difficult to verify, from (44) and (49), that (50) (51)which results in: (52)with (8) being the multiple characteristic root of the second-order ESO designed as in (43). Also, for plants of order , the resulting transfer function is given by (53) As can be noticed from the previous examples of lower order systems in (52) () and in (53) (), the formats assumed by the transfer functions revealed a well-defined mathematical pattern. Then, by extension, the generalisation for the case of systems of order n can be derived. Thus, one may expect that, for general n th order plants, the transfer function from to assumes the following format: (54) Remark 5.Here, it is important to mention that the adaption of the mixed representation for both time and frequency domain quantities, in (54), is only for analysis purpose. In the cases in which (38) involves general non-linear functions of the plant states, the definition of in the frequency domain may be not consistent. So, as it is intended to analyse the amplitude behaviour of the error signal in (54), without losing generality, we believe it is the most suitable input/output mathematical formalism to be utilised in this work. From the resulting expression in (54), it is possible to make a well-defined study about the amplitude in steady-state. From the theoretical point of view, the ESO (43) precision can be arbitrarily improved by increasing the absolute value of the characteristic roots at , which is the same effect of making the input/output equivalent gain to be very close to zero. An interesting property that is derived from (54) is the dynamical relation between the total disturbance term and its estimate , namely (55)which is obtained from (54), after replacing the following definition: (56) Inspired by the Bode diagram of (55), we define the following scaled relation between and : (57)where is introduced only for analysis purpose to represent a variable scaling factor, whose value belongs to the interval . Then, by rewriting (47), we have that (58) According to (36) and (38), the expression (58) can also be written as (59)Here, it is important to stress that the law of formation of in (59) involves a linear parametrised term in , which gives rise to a first order ODE. Recalling Assumption 3, the internal stability of (59) is guaranteed if is chosen as (60)and can be verified in the following (61) However, a more suitable way of analysing convergence and boundedness of system signals would be representing (58) using (33), (36) and (38), i.e. (62) For the sake of clarity, the dynamical equation in (62) is represented by the block diagram of Fig. 2. By rearranging the blocks of Fig. 2 in order to detach the signal as the diagram output and the signals as inputs, the diagram of Fig. 3 arises as a result. The polynomial that appears in Fig. 3 is given by (63)Then, by expanding (63), we have that (64)As can seen from (64), some of the coefficients of are linear combinations of the uncertain parameters of the original plant (23) and of the known filter coefficients , for . In order to ensure the stability of , in the presence of uncertain constants, we will choose (60) based on the constant upper and lower bounds and stated on Assumptions 2 and 3, respectively. Then (65)in which is a free design constant. By replacing (65) in some of the coefficients of in (64), we have that (66) (67) (68) Fig. 2Open in figure viewerPowerPoint Equivalent block diagram of (62) Fig. 3Open in figure viewerPowerPoint Rearranged block diagram of (62) From (66)–(68), we can highlight some important facts: the term is greater than the unity; The term is greater than any in (64); By choosing (65) as a sufficiently large constant, the products can be made larger enough for dominating every uncertain in (64). Then, by assuming is conveniently chosen, the polynomial can be approximated by (69) Moreover, if , then (69) can be simplified even more to reach the form, (70)which is guaranteed to be stable. The approximation in (70) is discussed in Lemma 1 in the following. Lemma 1.Consider a polynomial with and . If , then there always exists a real constant (71)such that the polynomial (72)can be approximated by (73) Proof.From (72), we know that (74) (75) (76) By expanding from (73), based on (74), we obtain (77)In (77), if is chosen sufficiently large, then dominant terms will arise inside the coefficients of for , i.e. (78)Since (78) must hold for every j and n, let us verify its viability. First of all, for the sake of simplicity, we assume that so that the values of elements in (76) have an increasing pattern . Thus, from (76) and according to the well-known Theorem of Binomial Coefficient Expansion, the three largest elements of will be given by (79)So, by normalising (79) by , we have that (80)which reveals the additional sufficient condition to be fulfiled. By expanding the normalisation done in (80) to the other coefficients in (76), it is not difficult to verify that is the largest scaling factor that can occur between any two of them. By considering the worst case of (80), we obtain a more general sufficient condition than (78), namely (81)Then, without loss of generality, the choices of , which can be suggested to satisfy (81) (and (78)), could be given by (82)Thus, once (81) is satisfied conveniently by (82), we can write, from (77) (), that (83)Then, (84)□ Remark 6.Note from the results of Lemma 1 applied to (66), that even without knowing exactly the values of the plant parameters (23), it is always possible to choose the design constant , based on the upper and lower bounds assumed for those parameters, in order to guarantee the stability of the polynomial . Here, it is important to mention that, although the stability of have been verified by using conservative assumptions about , Lemma 1 has established an algorithm for choosing it in a less conservative manner. Remark 7.The variable (57) introduced in the previous analysis is also an uncertain quantity. Although has influence in the choice of , it does not affect the stability of in (70), since it belongs to the interval . Moreover, it is worth to remember that, fo