# Robust Linear Output Regulation Using Extended State Observer

Authors: Mehran Hosseini-Pishrobat; Jafar Keighobadi; Atta Oveisi; Tamara Nestorović
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper presents a disturbance rejection-based solution to the problem of robust output regulation. The mismatch between the underlying plant and its nominal mathematical model is formulated by two disturbance classes. The first class is assumed to be generated by an autonomous linear system while for the second class no specific dynamical structure is considered. Accordingly, the robustness of the closed-loop system against the first disturbance class is achieved by following the internal model principle. On the other hand, in the framework of disturbance rejection control, an extended state observer (ESO) is designed to approximate and compensate for the second class, i.e., unstructured disturbances. As a result, the proposed output regulation method can deal with a vast range of uncertainties. Finally, the stability of the closed-loop system based on the proposed compound controller is carried out via Lyapunov and center manifold analyses, and some results on the robust output regulation are drawn. A representative simulation example is also presented to show the effectiveness of the control method.


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## Body

## 1. Introduction

The theory of output regulation in its essence deals with the problem of tracking/disturbance rejection of a class of signals generated by autonomous dynamic systems while guaranteeing the closed-loop stability. For linear systems, this theory is well-established, and the solution of the associated control problem is obtained by theinternal model principle[1, 2]. The principle is also instrumental in addressing the problem of robust output regulation. On this basis, robust design methods have been proposed in the literature for linear systems with uncertain parameters [3–6]. These methods consider the effect of external disturbances by an exogenous signal which belongs to the solution space of a particular differential equation. According to such a differential equation, the steady-state behavior of the underlying plant should be considered to examine the solvability of the output regulation problem. After determining the steady-state forms of the plant trajectories that satisfy the tracking/disturbance rejection requirements, the internal model-based design can be pursued [7]. By this method, a stabilizing controller that incorporates a suitable internal model of the class of the desired signals offers guaranteed asymptotic rejection/tracking of any signal from that class.In this paper, we depart from the conventional formulation of the output regulation problem by considering the effect of disturbances that cannot be modeled by a process signal of a differential equation. As a matter of fact, in many practical applications, such modeling assumption on the disturbances is restrictive. In many systems, the disturbances are not only caused by exogenous factors, but also from endogenous factors such as structural variations, parametric uncertainty, and unmodeled dynamics. The latter factors usually appear as state-dependent disturbances in the system’s mathematical model. For example, in many structural vibration control problems, the main source of the disturbance is the unmodeled higher order modes. Even though the dominant higher order modes can be obtained via suitable system identification methods, a comprehensive dynamic model for the disturbances is not feasible [8]. As another example, in many mechanical and electromechanical systems, parametric uncertainty, friction, and nonlinear effects bring about state-dependent disturbances that are not guaranteed to belong to the solution space of any particular dynamic system [9]. On this basis, our goal in this paper is to extend the applicability of the output regulation method to such examples. To this end, we recast the output regulation problem into the active disturbance rejection framework [10, 11]. Active disturbance rejection control (ADRC) is a robust control method for systems with large uncertainty and disturbances [10, 12–14]. The key idea of the ADRC is to treat a robust control problem as an estimation-rejection problem. More specifically, in the first step, all sorts of discrepancies between the physical system and its nominal mathematical model including parameter variations, unknown nonlinearities, and external disturbances/noises are lumped into atotal disturbance term [14]. Next, an estimator referred to as extended state observer (ESO) is applied to estimate the total disturbance and then reject it in a closed-loop framework. The active disturbance rejection control offers several promising advantages over the conventional control methods including the following: (1) ADRC is an active robustification method that in comparison to classical robust techniques (based on worst-case analysis) minimizes the conservatism in the design; (2) ADRC can handle large uncertainty/disturbances from both internal and external sources; (3) ADRC does not require involved, accurate system modeling; this feature enables a disturbance-oriented design instead of a model-oriented one [12–14]. More details about the ADRC and its various engineering applications can be found in [10, 14] and the recent book [13].In light of the above discussion, active disturbance rejection provides a suitable framework to generalize the applicability of the output regulation to the systems whose disturbances do not necessarily satisfy a differential equation. The main contribution of this paper is reported in terms of combining the merits of output regulation and disturbance rejection control techniques to achieve a better tracking performance in the presence of a general class of disturbance/uncertainty. The class of considered systems is fairly general, which includes linear systems subjected to mismatched disturbances as well as state-dependent uncertainties. Following the methodology of ADRC, we lump all disturbances/uncertainties of the underlying system into a total disturbance. The total disturbance term is decomposed into two parts. The first part can be classically modeled as the solution of a differential equation while the second part has no specific dynamical structure. The modeled component of the total disturbance is handled via standard method of the output regulation theory, which entails embedding an internal model into the closed-loop system. This approach will also guarantee output tracking of the reference signals that can be generated by the internal model. The unmodeled component of the total disturbance is handled by the method of active disturbance rejection. To this end, the system dynamics is extended by appending an integrator channel that transmits the uncertainty of the unmodeled total disturbance to its time derivative. Meanwhile, an ESO is designed that continuously monitors input-output signals of the extended system to estimate the unmodeled disturbance component. By combining this estimated value with the nominal output regulation control, the mismatch disturbance rejection is achieved. We note that since the ADRC uses an integral action to model the dynamical behavior of the disturbance, the method can be construed as an approximate output regulation technique [16]. This point of view is also explained in [17], where active disturbance rejection and output regulation are combined together for an improved robust vibration control of a MEMS gyroscope. As a novelty in the disturbance rejection aspect, a new ESO design method is proposed based on the convex programming. To this end, the ESO error dynamics is presented as a Lur'e system and its stability condition is phrased in terms of linear matrix inequalities (LMIs). Additionally, the proposed ESO applies nonlinear gain approach which enhances its immunity to the measurement noises.We note that various methods have been reported in the literature for the general problem of disturbance rejection/attenuation of control systems; for example, LMIs-basedL2 disturbance attenuation for nonlinear systems with input delays [18], operator-based disturbance attenuation/rejection [19], disturbance observer-based disturbance attenuation for stochastic systems [20], disturbance rejection based on the equivalent-input-disturbance approach [21], ESO-based disturbance rejection controller for fully actuated Euler-Lagrange systems [22], and sliding mode observer/controller hybrid control structure [15]. With respect to these methods, the main advantages of our method lie in combining the strong points of the output regulation and the disturbance rejection techniques. The output regulation component enables a guaranteed asymptotic rejection/tracking of the disturbances/reference signals with known dynamics. Moreover, for such disturbances/reference signals, no assumptions on the energy boundedness nor control matching are required. For disturbances without known dynamic behavior, the ESO-based disturbance rejection engages to fortify the output regulation. In other words, our method attempts to get the most out of the available information about the disturbances with the primary aim of disturbance rejection rather than disturbance attenuation.The remainder of this paper is organized as follows. In Section2, the control problem is formulated in terms of the output regulation theory. In Section 3, a nominal regulator is designed to handle the tracking problem as well as the rejection of the disturbance components with known dynamic characteristics. In Section 4, to compensate for the disturbances without known characteristics, an ESO is developed. A convex optimization-based design method for the ESO is proposed in Section 5. The stability of the closed-loop system is investigated in Section 6. In Section 7, a simulation example based on the plotter system is elaborated. Finally, the paper is concluded by Section 8.


## 2. Formulation of the Control Problem

Consider the following square state space model:(1)x˙t=Axt+But+γxt,t,yt=Cxt,where xt∈Rn is the state, ut∈Rm is the control input, yt∈Rm denotes the measured performance output, and γxt,t∈Rn×R+→Rn is a sufficiently smooth (differentiable) nonlinear function representing the total disturbance. A, B, and C are real matrices of appropriate dimensions satisfying the following assumption.Assumption 1.
The pair(A,B) is controllable and (C,A) is observable.The control goal is to asymptotically track a given reference signal, i.e.,ymt∈Rm, while boundedness of the all other signals is guaranteed in the presence of the total disturbance γxt,t. The reference signal ym(t) is considered to be generated by the solution of a fixed linear autonomous dynamic system of the form(2)w˙t=Swt,ymt=Cmwt,where wt∈Rr and S and Cm are real-valued matrices with proper dimensions. The following assumption is taken to guarantee the persistence and boundedness of the reference signal ymt [23].Assumption 2.
All eigenvalues of the matrixS have zero real parts with multiplicity one in the minimal polynomial.By settingσt≔γxt,t, the system in (1) can be rewritten as(3)x˙t=Axt+But+σt,yt=Cxt.The systems in (1) and (2) are trajectory equivalent in the sense that, under the same initial values, their state trajectories coincide with each other [24, 25]. Therefore, we carry out the controller design and analysis on the basis of (2).The conventional output regulation theory is based on the assumption that the perturbing disturbances belong to the solution space of the dynamic system given in (2). Here, we generalize this assumption by including disturbances which do not necessarily comply with this particular dynamics.Assumption 3.
The total disturbance signal satisfies(4)σt=Cσwt+Bξt,where Cσ∈Rr×r is a given matrix with the pair Cσ,S being observable. The signal ξt∈Rm is bounded and sufficiently differentiable such that ξ˙t is bounded as well.Remark 4.
The rationale behind Assumption3 can be explained as follows. In many engineering systems, only the dominant frequencies of the disturbances can be obtained via time and/or frequency domain identification methods (see, for example, [8]). In such systems, one can embed the known dominant frequencies into the system (2) and model the corresponding disturbance approximation error into the term, ξ.. As another example, consider the tracking control problems where the dominant dynamics of the underlying systems are linear. Assuming that the norm of the state-dependent disturbances is smaller than the convergence rate of the linear part, it is conceivable that, for a successful tracking, the frequency spectrum of the disturbances will contain the modes of the reference signals. Accordingly, (4) forms a suitable basis for modeling of the disturbances. An example of such modeling can be found in [9].
Let us define the tracking performance of the control system as(5)e≔y-ym.Note that here and after the dependence on time variable (t) is dropped (unless necessary) in the formulation for the sake of readability. The following composite system is proposed regarding the underlying output regulation control problem(6)x˙=Ax+Bu+Cσw+Bξ,w˙=Sw,e=Cx-Cmw.In order to examine the solvability of the output regulation problem, first, we consider a steady-state condition in which the output tracking is achieved. In such a steady-state condition, the trajectories of the underlying system should be well-defined to ensure that the solution of the output regulation problem exists. Denoting x-, u-, ξ-, and σ-≔γx-,t, as the steady-state vectors of the state, control, unstructured, and the structured disturbances, respectively, it is easy to obtain(7)x-˙=Ax-+Bu-+σ-,σ-=Cσw+Bξ-,Cx--Cmw=0.Setting the solutions as x-=Xw and u-=Uw-ξ-, for some X∈Rn×r and U∈Rm×r, we obtain the following Sylvester-type matrix equations from (7)(8)XS=AX+BU+Cσ,CX-Cm=0.Theorem 5.
For any given matricesCm and Cσ, there exist unique matrices X and U satisfying (8) if and only if the following assumption holds.Assumption 6.
For alls∈specS (spec. represents the vector of eigenvalues of the matrix (.));(9)rankA-sIBC0=n+m.Proof.
See [26] for the proof.From a geometrical point of view, solutions of the matrix equations (8) —known as theregulator equations—corresponds to a zero-error controlled invariant manifold defined by M≔x,w∈Rn×Rr∣x-Xw=0 for the composite system (6). In other terms, for an initial condition of the form x0=Xw0, under the control input u-, the trajectory x will evolve on M for all t∈R+ and the tracking error (5) will be identically equal to zero. From this standpoint, solving the formulated output regulation problem amounts to rendering the manifold M globally attractive by a suitable control function [7]. Since exact dynamic characterization of the total disturbance signal is not available, we pursuepractical stabilization of the manifold M for the composite system (6). More specifically, trajectories of the closed-loop system all should converge to a small compact set around M by approximately recovering the solutions of the regulator equations. To this end, we propose a disturbance rejection-based, two-degrees-of-freedom control structure. The control system is composed of a nominal output regulator equipped with an ESO for disturbance rejection. Accordingly, the main design stages of this control system are as follows.Stage 1.
The nominal controller solves the output regulation problem for the given dynamical system in (5) with the difference that the state equation is simplified to(10)x˙=Ax+Bun+Cσw.Note that the corresponding control input un∈Rm is designed to drive the nominal closed-loop trajectories toward M by recovering the nominal part of the steady-state control signal defined by u-n∶=Uw. This corresponds to the feedforward control input required for tracking ym as well as compensating for Cσw.Stage 2.
On the other hand, the disturbance rejection loop utilizes an ESO that continuously monitors the input-output signals of the plant to provide an estimate of the disturbance, denoted byξ^. The obtained estimated value is then integrated with the nominal control input to cancel out the disturbance signal. Accordingly, the overall control input is in the form(11)u=un-ξ^.The configuration of the proposed control system is schematically illustrated by the block diagram in Figure 1.Figure 1
Block diagram of the proposed control system.


## 3. Nominal Output Regulation

To solve the nominal output regulation problem, we characterize the distance of the system trajectories from the target manifoldM by a suitable coordinate. Then, by controlling this coordinate to zero, the output regulation will be accomplished. Considering the steady-state system trajectories on the manifold, an intuitive candidate for such a coordinate is x-x-. However, owing to the direct dependence of the regulator equations on the solutions of ξ^, this method is sensitive to perturbation (see [7]). Consequently, the intuitive candidate may render the control system vulnerable to the disturbance estimation errors. Hence, we follow a more robust design approach based on the internal model principle. For this purpose, setting p∶=d/dt, the following linear differential operator is considered:(12)Γp=pl+∑i=0l-1αl-i-1pl-i-1,where αi|i=0l-1 with αl≔1 are the coefficients of the minimal polynomial of the matrix S in the ascending order. Next, we introduce the following transformed variable:(13)xt=Γpx.Proposition 7.
x t ≡ 0, if and only if (x,w)∈M.Proof.
The motion of the system trajectories onM under the control input u-n is equivalent to the immersion of system (10) into (2). By properties of the minimal polynomial, for any trajectory of w in (2), we have Γpw=0. Thereby, the set of state trajectories evolving on the manifold M corresponds to the kernel of the operator Γp; i.e., KerΓ.=x:R+→Rn∣Γ.x=0. Thus, the variable xt acts as an indicator function for the set KerΓ. in the sense that a state trajectory xt evolves on M if and only if its transformation xtt is equivalent to zero.In view of Proposition7, the variable xt is a distance coordinate characterizing attractivity of the manifold M for trajectories of system (10). By embedding the internal model of the reference signal, this distance coordinate does not depend on the solutions of the regulator equation and, therefore, enables a robust design. Interestingly, based on this argument, the output regulation problem entails the problem of regulating xt to zero. Hence, by applying the operator (12) to the nominal state equation (10), we obtain the following dynamics: (14)x˙t=Axt+But,in which the following control transformation is introduced:(15)ut≔Γpun.Since the variable xt is not available for feedback, system (14) cannot be stabilized by the standard methods. To overcome this issue, we obtain the following differential equation describing the dynamic relation between the distance coordinate and the reference tracking through applying Γp to the tracking error (5):(16)el+∑i=0l-1αl-i-1el-i-1=Cxt.This differential equation is converted to the following state space representation simply by considering xt as an input:(17)x˙e=Λxe+ECxt,e=Gxe,in which xe∶=cole,…,el-1 and(18)Λ=0I…000⋯0⋮⋮⋱I-α0I-α1I…-αl-1I,E=0⋮I,G=I0…0.Setting xaug∶=colxt,xe, while selecting e as the fictitious output, both (14) and (17) are integrated into the following augmented system:(19)x˙aug=Aaugxaug+Baugut,e=Caugxaug,where(20)Aaug=A0ECΛ,Baug=B0,Caug=0G.Proposition 8.
Under Assumptions1 and 6, the augmented system (19) is both controllable and observable.Proof.
According to the Popov-Belevitch-Hautus (PBH) test, the following rank conditions should be verified for alls∈specAaug; i.e.,(21)ranksI-AaugBaug=n+ml,(22)ranksI-AaugCaug=n+ml.Controllability. To assure the controllability of the augmented system (19), the rank condition (21) should hold. By expanding the corresponding matrix and after an appropriate rearrangement of the block rows and columns, the rank condition is inherited over (23)rankBsI-A00…00-Cα0Iα1I…s+αl-1I00sI-I…0⋮⋮⋮⋮⋱⋮000…sI-I.By the controllability condition of Assumption 1, the first block row has the rank of n for all s∈C. Consequently, owing to the identity matrices with different column indices, the last l-1 block rows have the rank ml-1. Note that the second block row is independent of the other rows, unless for s∈specS. Thereby, to guarantee the controllability, the following rank condition needs to be satisfied for all s∈specS:(24)rankBsI-A0-C=n+m.Clearly, in view of Assumption 6, this condition is verified.
Observability. To show observability of the augmented system (17), the rank condition (22) needs to be verified. Again, we expand the pertinent matrix rank condition as (25)ranksI-A00…00sI-I…0⋮⋮⋮⋱-ICα0Iα1I…s+αl-1I0I0…0.Due to the observability condition of Assumption 1, the first block column has the rank n for all s∈C. The last l block columns are independent due to the identity matrices with different row indices. Accordingly, the overall rank of the matrix is n+ml. This completes the proof.In order to stabilize the augmented system (19), we propose a dynamic output feedback controller of the form(26)x˙κ=F1xκ+F2e,ut=F3xκ,where xκ∈Rnκ is the internal state of the controller and Fi,i=1,2,3 are real matrices of appropriate dimensions. The controller (26) in conjunction with the augmented system (19) yields a closed-loop system governed by the following equation:(27)x˙s=Asxs.Here xs∶=colxaug,xκ is the aggregated state vector. Moreover, the system matrix is given as(28)As=AaugBaugF3F2CaugF1.Owing to Proposition 8, there always exist matrices F1, F2, and F3 such that the resultant closed-loop system is exponentially stable. In view of this observation, without loss of generality, the following assumption is made.Assumption 9.
The controller (26) is synthesized in such a way that the matrix As is Hurwitz stable.
The proposed output regulation method uses the transformed control variable (15) in its feedback loop, i.e., ut. Consequently, the primary control input is readily available by applying the inverse of the operator Γp to the transformed variable. To this end, the following state space realization is employed:(29)x˙u=Λxu+Eut,u=Gxu,where xu∶=colu,…,ul-1 and the system matrices are the same as those of (17). The controller (26) together with the system (29) constitute the nominal output regulation loop of the overall control system.Remark 10.
System (29) replicate a model of the exogenous signal, w, defined by (2). Therefore, the proposed nominal output regulation has the internal model property which, in turn, guarantees asymptotic rejection/tracking of any signal belonging to the solution space of (2) [7].


## 4. Extended State Observer

The output regulation technique presented in Section3 can handle the disturbances whose frequency spectra are embedded in the modes of system (2). For compensation of the disturbances without known frequency characteristics—represented by the signal ξt—an ESO is designed. Note that, trivially, based on the matched condition, the unstructured disturbance is assumed to belong to the range space of B; otherwise the transfer function from disturbance to the output of interest cannot be set to null (see the geometrical approach proposed in [27]). Considering the composite system (6), the following ESO is proposed:(30)x^˙=Ax^+Bu+Cσw^+Bξ^+H1ϕy-y^,w^˙=Sw^+H2ϕy-y^,ξ^˙=H3ϕy-y^,where x^, w^, and ξ^ are the estimates of the states, structured, and unstructured disturbances, respectively. Additionally, the estimated system output is y^=Cx^. In ESO dynamics, Hi,i=1,2,3, are design matrices of appropriate dimensions. Based on the nonlinear observer proposed by Prasov and Khalil [28], for positive scalars 0<ε<1, and d>0, the nonlinear gain function of the observer, i.e., ϕ., is defined as(31)ϕz=ϕ1z1…ϕmzmT,ϕjzj≔εzj,zj≤dzj+dε-1signzj,zj>d,j=1,…,mRemark 11.
The gain function (31) is a dead-zone nonlinearity decreasing the observer gain when the estimation error falls inside the zone [-d,d]. This allows adjusting a trade-off between fast state estimation and robustness to measurement noises.
In order to investigate convergence of the ESO (30), defining the error variables as η1∶=x-x^, η2∶=w-w^, and η3≔ξ-ξ^, we obtain the following differential equation for the estimation error dynamics:(32)η˙=A0η+Hϕς+Qξ˙,ς=C0η,where η∶=colη1,η2,η3 and the system matrices are given by(33)A0=ACσB0S0000,H=-H1-H2-H3,Q=00I,C0=C00.Assumption 12.
The unperturbed error dynamics, given in(34)η˙=A0η+Hϕς,ς=C0η,has a globally exponentially stable equilibrium at the origin. The feasibility of this assumption is addressed in Section 5.Theorem 13.
Under Assumptions3 and 12, the estimation error of the ESO (30) is globally bounded and globally ultimately bounded in the sense that, after a finite transient time T0>0, the following bound is valid:(35)η≤ρ0+βξ0,where ρ0>0 is arbitrarily small while β>0, and ξ0≔supt≥0,⁡ξt.Proof.
By Assumption12, there exist a positive definite function Vη and positive scalars β0, β1, and β2 satisfying (see Theorem 5.17 of [29]) (36)dVηdt32≤-β0Vη,∂Vη∂η≤β1Vη,η≤β2Vη.Calculating the time derivative of Vη along the solution of (32) gives (dVη/dt)(31)=(dVη/dt)|(32)+(∂Vη/∂η)Qξ˙≤-β0Vη+β1ξ0Vη. By dividing both sides of the last inequality by 2Vη, invoking the comparison lemma and using (36), we obtain η≤β2Vη-(β1/β0)ξ0exp⁡-β0t/2+β1β2ξ0/β0, which clearly shows the global boundedness of the estimation error. Moreover, considering the decaying exponential term, after the time T0=(2/β0)Logβ2Vη-(β1/β0)ξ0/ρ0, the inequality (35) holds with β≔β1β2/β0.


## 5. ESO Design via LMIs

The key to the convergence result of Theorem13 is Assumption 12, which requires the existence of a positive definite function Vη satisfying (36). In this regard, we propose a convex programming-based method to construct the function Vη and also to design the gain matrices Hi,i=1,2,3 in (29). Instrumental for the development of our method, the unperturbed error dynamics (34) is considered as a Lur'e system in terms of the gain function ϕς (which is assumed to be a sector nonlinearity).Remark 14.
The gain functionϕ. belongs to the sector [1,ε]. That is, for all ς∈Rm,(37)εςTς≤ςTϕς≤ςTς.Using (34), the sector condition (37) is equivalent to the following quadratic inequality (see [30]):(38)εηTC0TC0η+ϕTςϕς-1+εηTC0Tϕς≤0.In the framework of quadratic stabilization, we search for a positive definite function of the form Vη=ηTPη, which guarantees Assumption 12 by satisfying the first requirement of (36). Thereby,(39)ηTPA0+A0TP+β0Pη+2ηTPHϕς≤0.We should note that since Vη is a quadratic function, the other two requirements of (36) are trivially satisfied according to Rayleigh’s principle.Theorem 15.
For a given constantβ0>0, assume there exists a positive definite matrix P, a matrix Y with appropriate dimension, and a scalar ϑ≥0, satisfying the LMI(40)A0+A0TP+β0P-εϑC0TC0Y+ϑ1+ε2C0TYT+ϑ1+ε2C0-ϑI≤0.Then, the function Vη, along with the gain matrix H=P-1Y, guarantees the exponential stability of the origin of (34).Proof.
The origin of the system (34) is exponentially stable if the inequality (39) holds for all η and ϕς satisfying the sector condition (38). According to the S-procedure Lemma [30], this statement is equivalent to the existence of a scalar ϑ≥0 such that ηTPA0+A0TP+β0Pη+2ηTPHϕς-ϑεηTC0TC0η+ϕTςϕς-1+εηTC0Tϕς≤0. Convexifying the inequality by the transformation Y=PH, and using the Schur complement Lemma, LMI (40) is achieved. Thus the proof is complete.


## 6. Closed-Loop Stability

The modular structure of the proposed control system enables an independent design of the output regulation and disturbance rejection loops. However, due to the interaction of the components, the overall closed-loop system stability still needs to be investigated. The stability of the disturbance rejection loop is established by the ESO convergence results of Theorem13 and the design method presented in Theorem 15. Therefore, we consider the effect of the ESO-based disturbance rejection on the performance of the nominal regulator. The nominal closed-loop system operating in conjunction with the disturbance rejection loop is described by the equations of the form(41)w˙=Sw,x˙cl=Aclxcl+Bclw+Qclξ-ξ^,where xcl∶=colx,xu,xκ, and(42)Acl=ABG00ΛEF3F2C0F1,Bcl=Cσ0-F2Cm,Qcl=B00.Additionally, to examine the effect of disturbance rejection on regulating the distance coordinate xt, we consider system (27) perturbed by the disturbance cancellation error(43)x˙s=Asxs+Qsξt,where ξt≔Γpξ-ξ^ is the transformed perturbation and Qs=BaugT0T.Assumption 16.
The signalξt. is Lebesgue measurable and bounded.
The next lemma is instrumental for the closed-loop stability analysis.Lemma 17.
We havespecAs=spec(Acl).Proof.
We show that the characteristic polynomials ofAs and Acl coincide. For this, it suffices to show that for all s∈C(44)det⁡As-sI=det⁡Acl-sI.First, regarding the block components of As, we have(45)det⁡As-sI=det⁡A-sI0BF3ECΛ-sI00F2GF1-sI=detA-sIdet⁡Λ-sI-ECA-sI-1BF3F2GF1-sI=detA-sIdetΛ-sIdet⁡F1-sI+F2GΛ-sI-1ECA-sI-1BF3.Following the same procedure for Acl, we obtain(46)det⁡Acl-sI=detA-sIdetΛ-sIdet⁡F1-sI+F2CA-sI-1BGΛ-sI-1EF3.Since GΛ-sI-1E=-(1/Γs)I, the matrix multiplication of GΛ-sI-1E commutes with CA-sI-1B and, thereby, equality (44) holds for all s∈C.Theorem 18.
Consider the system (41). Assume that Assumptions 1–16 are satisfied, λ0∶=-maxs∈specAcl⁡Res, and ξt0∶=supτ≥0⁡ξtτ. Then,(1)
The state trajectoriesx, xu, and xκ are all globally bounded.(2)
In the state space of the composite system (6), the state trajectory x converges practically to the manifold M in the sense that(47)x-x-≤μ0′+μ1′exp⁡-λ0t,for some positive scalarsμ0′ and μ1′.(3)
The error variablee and its first l-1 derivatives are globally bounded and globally ultimately bounded. That is, there is a finite time T0′′>0, after which the following inequalities holds:(48)eit≤ρ0′′+μ1′′ξt0λ0,i=0,…,l-1,whereρ0′′>0 is arbitrarily small and λ0>0 is a pertinent constant.Proof.
The proof is separated into three parts.
(I) We consider the closed-loop equation (41) without perturbation(49)w˙=Sw,x˙cl=Aclxcl+Bclw.In view of Lemma 17 and Assumption 9, the matrix Acl is Hurwitz stable. Therefore, according to the center manifold theorem [29], there exists a stable invariant manifold M′ attracting all trajectories of (49). The motion of the system trajectories on M′ is equivalent to the immersion of the system (49) into (2). Hence, there is a linear map of the form xcl=Πw, in which the matrix Π satisfies the Sylvester equation(50)ΠS=AclΠ+Bcl.We note that existence of Π is guaranteed since specS∩spec(Acl)=∅. For the perturbed closed-loop dynamics (41), an attractivity index with respect to M′ is defined as q∶=xcl-Πw. Then, it is straightforward to obtain(51)q˙=Aclq+Qclξ-ξ^.By Theorem 13 and following the same notation, the disturbance estimation error is bounded as ξ-ξ^≤βξ0+η0exp⁡-β0t/2, for some η0>0. As a result, the solution of (51) satisfies an inequality of the form(52)q≤βξ0μ0+μ1exp⁡-λ0t+μ2exp⁡-β0t2,where μ0, μ1, and μ2 are appropriate positive numbers. Inequality (52) implies practical convergence of the closed-loop trajectories to the manifold M′. Using the triangle inequality xcl≤q+Πw, boundedness of the trajectory xcl is deduced.
(II) Apportioning matrixΠ as(53)xxuxκ=ΠxΠuΠκw,the following matrix equation is obtained from (50):(54)ΠxS=AΠx+BGΠu+Cσ.Comparing (54) with (8), it follows from the uniqueness result of Theorem 5 that X=Πx, and U=GΠu. Thereby, M is a submanifold of M′, and finally it is implied from inequality (52) that x-x-≤μ0′+μ1′exp⁡-λ0t, in which μ0′=βξ0μ0+μ2, and μ1′=μ1.
(III) The solution of the state equation (43) satisfies xst≤xs0-Qsξt0/λ0exp⁡-λ0t+Qs(ξt0/λ0). Given ρ0′′>0, after the transient time T0′′∶=(1/λ0)Logxs0-Qsξt0/λ0/ρ0′′, the inequality xst≤ρ0′′+μ1′′(ξt0/λ0) holds with μ1′′=Qs. This completes the proof.


## 7. Simulation Example

In order to illustrate the efficacy of the proposed control method, we explain its application to the plotter system, which is a computer printer used for vector graphics [15]. The schematic model of the plotter is shown in Figure 2, and the corresponding symbols and numerical values are given in Table 1. The basic components of the plotter are a DC motor and three disks (referred to as disks 1-3). The state variables of the system x1, x2, and x3 are the angular displacement of disk 1, the angular velocity of disk 1, and the current in the motor, respectively. The control variable is u in the input voltage of the motor. Through a first principle modeling, we obtain the state equation of the plotter as follows:(55)x˙=Ax+Bu+γx,t,where(56)
A
=
0
1
0
-
K
1
+
r
1
/
r
2
2
K
2
J
1
+
r
1
/
r
2
2
J
2
+
1
/
2
M
r
2
2
-
B
1
+
r
1
/
r
2
2
B
2
J
1
+
r
1
/
r
2
2
J
2
+
1
/
2
M
r
2
2
K
i
J
1
+
r
1
/
r
2
2
J
2
+
1
/
2
M
r
2
2
0
-
K
v
L
m
-
R
m
L
m
,
B
=
0
0
1
L
m
.The total disturbance comprises two terms in such a way that γx,t=γ1t+Bγ2(x). The first, which has a known frequency spectrum, affects both motor dynamics and mechanical part of the system and is given by γ1t=0,0.01sin⁡20t,0.2sin⁡(20t)⊤. The second term is a matched state-dependent uncertainty of the form γ2x=-0.1tanh⁡(10x2), which represents frictional effects in the motor. Taking the disk 1 angle as the output, y=x1, we assume that the control objective is to track a reference signal of the form, ymt=0.01sin⁡t. According to the available frequency data of the reference/disturbance signals, system (2) is characterized by the following matrices:(57)S=0100-1000000100-4000,Cm=1000.In view of Assumption 3 and Remark 4, we consider the disturbance model (4) with(58)Cσ=000010101010.The controller (26) is designed using linear quadratic regulation in such a way that all eigenvalues of the matrix As are placed on the left of the line R(s)=-3 in the complex plane. Parameters of the nonlinear gain function (31) are selected as ε=0.8 and d=0.1. To obtain the matrices Hi,i=1,2,3, taking β0=0.01, the LMI (40) is solved per CVX [31]. We obtained(59)H1=32.598626.66380.5471,H2=14.0456-4.1513-7.9373-442.8252,H3=35.6611.We also compare the tracking response of the controller with the hybrid controller [15], which comprises a sliding mode observer and a sliding mode output feedback controller. Since only the stabilization problem is considered in [15], we perform the following state and control transformation:(60)x-1=x1-ym,x-2=x2-y˙m,x-3=x3-θm,u-=uLm+a32y˙m+a33θm-θ˙m,θm=-1a23a21ym+a22y˙m-y¨m,where aij are the entries of the state matrix A. According to the transformation (60), the following system is obtained for the stabilization problem:(61)x-˙=Ax-+Bu-+γ2t+γ1t,e=Cx-,where x-=x-1,x-2,x-3⊤ and C=100. The hybrid controller is given by(62)x-^˙=Ax-^+Bu-+Khybe-Cx-^+NhybsignMhybe-Cx-^,γ^2=B+Khybe-Cx-^+B+NhybsignMhybe-Cx-^,u-=-γ^2-B+Khybe+B+NhybsignMhybe,where x-^ and γ^2 are estimates of x- and γ2, respectively, Khyb, Nhyb, and Mhyb are design matrices of suitable dimensions (MhybC should be positive semidefinite), and B+ is the pseudoinverse of B and is the standard signum function [15]. We note that in the design of the hybrid controller only matched disturbances are considered [15]. Using eigenvalue placement, the design matrices of the hybrid controller are tuned as follows:(63)Khyb=0.0675,0.8936,1.4613⊤×103,Nhyb=0.01,Mhyb=1,1,1⊤.The tracking responses of both controllers are shown in Figure 3. The settling times of the proposed controller and the hybrid controller are 4s and 25s, respectively. Our controller shows a much faster transient response, but its overshoot is larger. The approximate values of the absolute steady-state tracking errors of the controllers are 1.567×10-7rad for the proposed controller and 1.219×10-4rad for the hybrid controller. That is, our controller outperforms the hybrid controller in terms of transient time and steady-state tracking. This is mainly due to the output regulation property of our controller which enables guaranteed asymptotic tracking as well as mismatched disturbance rejection. The control signals generated by the controllers are shown in Figure 4. Both control signals are approximately the same; however, there is a high-frequency component in the control input of our controller. This component is caused by the internal model property of our controller which embeds the known frequencies of the reference/disturbances to learn their behavior [8]. In order to assess the performance of the ESO, the estimated state variables of the plotter system are given in Figure 5. According to the ESO differential equations (30), state reconstruction is a part of disturbance estimation. The ESO successfully tracks the state variables of the system, which, in turn, confirms its disturbance estimation capability.Table 1
Symbols of the plotter system model with the numerical values [15].
Symbol Description Value K i Motor magnetic flux 10 - 3  k g . m 2 / s 2 K v Motor back electromotive force 0.45  V . s / r a d R m Motor resistance 1  Ω L m Motor inductance 10 - 2  H J 1 Disk 1 moment of inertia 0.01213  k g . m 2 B 1 Disk 1 viscous friction 0.1  N . m . s / r a d K 1 Disk 1 elastic constant 10 - 3  N . m / r a d r 1 Disk 1 radii 0.1  m J 2 Disk 2 moment of inertia 0.01213  k g . m 2 B 2 Disk 2 viscous friction 0.1  N . m . s / r a d K 2 Disk 2 elastic constant 10 - 3  N . m / r a d r 2 Disk 2 radii 0.1  m M Disk 3 (the pen nib) mass 0.5  k gFigure 2
Schematics of the plotter system [15].Figure 3
Tracking responses of the proposed controller and the hybrid controller.Figure 4
Time trajectories of the control inputs of the proposed controller and the hybrid controller.Figure 5
Estimation of the plotter state variables using the ESO.


## 8. Conclusion

This paper presented a control structure combining output regulation and disturbance rejection control techniques. This method extends the range of the disturbances considered in the conventional output regulation theory. In the output regulation aspect, the internal model principle enables the controller to handle the mismatched disturbances with known frequency characteristics. In the disturbance rejection aspect, an ESO is employed to compensate for the disturbances without known dynamical characteristics, which, for example, may arise from nonlinear effects and structural variations of the underlying plant. Generalization and application of the method to a class of mechanical systems will be pursued in future works.



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*Source: 4095473-2018-08-12.xml*