Hebertt Sira‑Ramírez · Mario Andres Aguilar‑Orduña · Brian C. Gómez‑León

Abstract

Keywords Active disturbance rejection control · Lagrangian systems · Robust control · Flatness in Lagrangian systems

1 Introduction

In this article, a direct tangent linearization procedure is presented for nonlinear SISO controlled Lagrangian systems on the basis of a second order expansion, around a given equilibrium point, of the Lagrangian function. When the tangent linearization of the nonlinear Lagrangian system is controllable, then an incremental flat output exists, modulo a scalar constant factor. The nonlinear system may then be controlled on the basis of its tangent linearization using a robust scheme with respect to neglected nonlinearities and other unmodelled disturbances. For linearized controllable Lagrangian systems, the incremental flat output is shown to be a linear combination of the incremental generalized position variables. Moreover, the input-to-flat output dynamics representation of the linearized system entitles only even order time derivatives of the incremental flat output. These time derivatives also depend, solely, on generalized position variables. These two facts enormously facilitate the feedback controller design for flat output reference trajectory tracking task thanks to the possibilities of precise higher order time derivatives cancellations, via use of generalized position variables for feedback. The nonlinear system controller design task may be reduced to that corresponding to a pure integration system.

Adding to the above facts, the ROESO based ADRC controller design is further facilitated thanks to its equivalence with classical compensation networks, and GPI controllers,expressed in transfer function form. The ROESO based ADRC controller is shown to be easily obtainable in terms of endogenous injections and exogenous feedback (EIEF),requiring only the output tracking error and the control input error for robust dynamic feedback purposes.

Second order variational expansions of Lagrangian functions, for small signal model derivations, were suggested by V. I. Arnold in his seminal Classical Mechanics book [1] (see also Crouch and van der Schaft [2] for closely related use of adjoint variational systems). The relations between ADRC and classical compensation networks, as developed in [3], is invoked for proposing a classical, frequency domain, linear solution of the underlying nonlinear tracking problem. Clearly the excited nonlinearities, activated by the system (fast) tracking maneuvers, may be substantially attenuated using ADRC,or any equivalent technique. This design route is equivalent to approximate extended state observer-based endogenous disturbance estimation and ulterior feedback cancellation as in traditional ADRC schemes (see Han [4], Han and Gao [5]).For a closely related approach the reader is referred to the work of Xue and Huang in [6]. ADRC control of nonlinear Lagrangian, or Hamiltonian, systems, with controllable tangent linearizations, were presented in [7] for multivariable Lagrangian systems.

The equivalence between ROESO based ADRC and Classical Compensation Networks (such as GPI control) is shown to be valid under a stable co-prime factorization of the closedloop system characteristic polynomial, induced by the separation principle.

Section 2 considers the direct linearization of a Lagrangian System via the variational incremental expansion of the Lagrangian function and the detailed study of controllable linearized Lagrangian systems. The controller design task reduction, to that of a pure integration system, motivates the study of classical compensation networks via a straightforward EIEF approach in frequency domain terms. Surprisingly, in this classical form, an equivalence clearly arises between GPI control and ROESO based ADRC control, provided the separation principle is duly respected. These findings are detailed in Sect. 3. Section 4 is devoted to an illustrative example including an output reference trajectory tracking task on a rotary underactuated crane model (which is non-feedback linearizable, i.e., non flat). The performance of the synthesized linear controller is thoroughly tested on the nonlinear system for output reference trajectories causing large enough deviations from the operating equilibrium point. Section 5 contains the conclusions and suggestions for further research.

2 Tangent linearization of controlled Lagrangian systems

Define the scalar function, L(q,˙q,u) , to be the Lagrangian of a controlled system,q∈ℝn,u∈ℝ , given by

where K(q,˙q) is the kinetic energy associated with the motions of the masses and the rotations of inertias. The term,V(q)-qTbu, represents the total potential energy comprising a) the potential energy V , presented by elastic and gravitational effects, and b) the term -qTbu, representing the works,exercised by the control input,u, on the system, through the input channel represented by the constant vector,b.

where it is assumed that the matrix M(q)∈ℝn×nis an invertible, symmetric, matrix.

Define the incremental variables:qδ,˙qδ,uδ, as the variational deviations from the equilibrium point of the position,velocity and control input vectors.This implies that all linear terms, in the expansion (6), disappear at the equilibrium, except for that associated with the incremental control input vectoruδ. In equilibrium, the elastic forces and the gravitational forces are in the image of the constant mapband they are counteracted by the equilibrium control inputs

The tangent linearization procedure and the determination of the approximate linearized dynamics, in terms of the incremental Lagrangian, follows from the following identities:

The incremental, linearized, Euler-Lagrange equations are obtained directly from the expression isomorphic to the traditional formalism

2.1 Example

Consider the case of asimple, controlled, Lagrangian system, defined by (3). In equilibrium: (,0,) , the Lagrangian is reduced to

2.2 Flatness and the incremental Lagrangian system

The first block matrix is full rank. It is evident, then, that the upper block ofn-vectors in the second matrix, as well as the lower block ofn-vectors, (which has the same rank as the first block), are full rank. Therefore, the pair: (-KM-1,b) ,is necessarily controllable, i.e.,

is full rankn. It is clear that if the pair, (KM-1,b) , is controllable, the composite matrix is full rank, 2n, and, hence,the overall system is controllable.

The minus signs, in front of the matrices: (KM-1)j, do not affect the rank of (18). We have, under the above assumptions,that the controllability condition for the overall system may be stated in terms of the full ranknof the matrixKgiven by

Therefore, the controllability matrix (17) can be written as the 2nfull rank matrix product,and clearly, the components of the incremental flat outputyδdepend only on the position coordinatesqδ:

It is clear from (12) that, in general, the even order time derivatives of the components of the incremental flat output vector,yδ, depend only on the components of the vector of incremental generalized positions,qδ, while the odd order time derivatives of,yδ, depend only on the components of the incremental generalized velocities,ϑδ.

The incremental input to incremental flat output, linear,relation is therefore of the form:

whereξ(t) depends on even order time derivatives of the flat outputyδ, and, possibly, on unmodelled exogenous disturbances.

Equation (26) as a generic model for nonlinear SISO Lagrangian systems with flat, or controllable, linearizations motivate the classical controller design approach undertaken in the next section. In so doing we find an equivalence of EIEF based controllers with classical compensation networks, and more interesting, with ROESO based ADRC controllers. To simplify the treatment we usenas the order of the flat system rather than 2n. The translation of the results for Lagrangian systems is straightforward.

3 Equivalence of EIEF and ROESO‑ADRC in pure integration systems

Consider a, perturbed, pure integration system of the form

It is desired to have the measured output of the nonlinear system,y, track a given, smooth, output reference signal,y*(t) , devising a suitable feedback control input law,u,without measurements of output variable times derivatives,i.e., ofy(j),j=1,…,n-1 , and with no knowledge of the

3.1 A flat filter (FF) controller and its advantageous features

We adopt, henceforth, the terminology used in the book by Åström and Murray [10], in reference to the classical block diagram in Fig. 1.

Given the perturbed tracking error system (30), a classical compensator network controller of the form,

yields the closed-loop system:

Fig. 1 Classical block diagram

withω′n=ωn∕∈and 0＜∈＜1.

Fig. 2 Bode Magnitude plot for the Open Loop Gain transfer function

Typically, output reference signals are low frequency signals. Input disturbances are also low frequency signals.Measurement noises are predominantly high frequency signals. The Bode magnitude of the open loop gain transfer functionG(s)K(s), and those corresponding to the “gang of four”, reveal: desirable output reference trajectory tracking accuracy, good high frequency output measurement noise rejection, significant input disturbance attenuation at all frequencies, being large at very low and very high frequencies.High frequency measurement noise magnification at the control input is upper bounded while output reference signals do not significantly contribute to control input magnitude at low frequencies but, rather, at mid and large frequencies.The proposed controller satisfies the customary practical requirements for a good closed-loop performance. It is a common task to make the linearized plant behave as close as possible to a pure integration plant by cancelling all influences of external disturbances and those of the neglected non-linearities. These are excited on the nonlinear system by the reference trajectory tracking features which include taking the system operation away from the equilibrium point used in the tangent linearization. This is the main feature of observer based ADRC schemes (see [5] and [11]). It is not difficult to see that, due to their specific nature, none of the transfer functions in the “gang of four” admit an unstable pole zero-cancellation, while the closed-loop characteristic polynomial is purposefully designed to be Hurwitz. The closed-loop system is, therefore, asymptotically stable.

3.2 An EIEF controller in classical transfer function form

Consider the auxiliary, state space, extended tracking error system, including endogenous injection and exogenous feedback subsystems:

Fig. 3 Bode Magnitude plots of the “gang of four” transfer functions for integration plant and classical compensation network controller

3.3 A ROESO based ADRC controller in classical form

Consider the pure integration, additively perturbed, tracking error input-outputnth order system

We adopt, for the derivation of the reduced order observer,then-1 dimensional (plant) system state representation,including an artificial velocity tracking error measurement(y2∶=˙y-˙y*(t)=∶˙y1=˙ey). We clarify that this is an auxiliary procedure and that no output time derivative will be needed in the final form of the controller. Denoteey=∶y1.

which implies large attenuation, at low frequencies, for the effect of the input disturbance signal,ξ(t) , in the state and disturbance estimation errors produced by the proposed ROESO.

Finally, use=y1=ey. A ROESO-based ADRC controller, for output stabilization purposes, is proposed as

Letpo(s) denote the observer’s characteristic polynomial:po(s)=sn+λn-1sn-1+…+λ1s+λ0and letpc(s) denote the closed-loop characteristic polynomialpc(s)=sn+γn-1sn-1+…+γ1s+γ0. Using the above expressions (54), one obtains, in the Laplace transform domain:Thus, given a ROESO based ADRC controller, through its coefficientsγ’s andλ’s, it takes a direct computation to uniquely obtain the 2ncoefficients,kj,j=0,…,2n-1 , of the classical compensation network. Denote byKthe 2ncolumn vector of components {k2n-1,…,k1,k0} vertically arranged in descending order. For the reverse computation,i.e., to compute theγ’s andλ’s, given the vector of coefficients,K, of a classical compensation network, we have to solve the set of quadratic, 2n, equations in 2nunknowns.Denoting, respectively, byΛandΓthen, column, vectors[γ0γ1…γn-1]Tand [λ0λ1…λn-1]T, we have

The load sensitivity transfer function,GS(s), for the closedloop ROESO-ADRC controlled system and the controller,βK(s) , entirely coincide with those derived via EIEF (or GPI) control, and they also coincide with those found for the FF classical compensator network.

The characteristic polynomial of the closed-loop system factors into the product of the ROESO characteristic polynomial (i.e., withn=(n-1)+1 ) and thenth order characteristic polynomial of the closed-loop system, obtained by straightforward pole placement on thenth order pure integration plant system, as if all the output phase variables had been available for feedback. All this is, evidently, in accordance with the observer-controller stable design separation principle for state feedback control through a state observer in the realm of linear controllable and observable systems.

Clearly, given a ROESO-based ADRC controller design,there exists a unique stable flat filter, or GPI, controller which has exactly the same set of fundamental transfer functions (sensitivity, complementary sensitivity and open loop transfer functions). On the other hand, given a GPI (FF) controller design, there exists non-unique equivalent ROESO based ADRC controllers. The nonlinear map, relating the coefficients of the closed-loop characteristic polynomial obtained for a given GPI controller, with those of the corresponding ROESO based ADRC is given by (γn,λn=1),

computed at the solution (,) ofg(Γ,Λ)=.

The Jacobian obtained coincides, module a reordering of the matrix columns, with the “eliminant” matrix, also known as the Sylvester matrix, customarily used in coprimeness tests of two given polynomials.

The Sylvester matrix is simply given by the following matrix:

The two 2n×nsubmatrix blocks are isomorphic. Invertibility of thegmap rests on the assumption that the observer and the controller have non-identical coefficients in their characteristic polynomials and, hence, that the location of their poles are not coincident.

The Sylvester matrix (66) is full rank, if and only if the monic polynomials,po(s) andpc(s) , are co-prime. We have then: Given the coefficients {k2n-1,k2n-1,…,k1,k0} of a designed monic GPI (FF) closed loop ch.p., the equivalent ROESO-ADRC observer-controller design coefficients can be uniquely computed, provided the 2nth order closed loop ch.p. of the GPI controlled system is factored, in a co-prime manner, into twonth order monic polynomials.

The equivalence just established has been also tested in several experimental settings (see [13, 14]). It is substantially helpful in synthesizing observer based ADRC control schemes for nonlinear systems. In our setting it adopts the form of a stable proper transfer function with enhanced disturbance attenuation features and good low frequency output reference trajectory tracking features.

4 A controlled underactuated rotary crane system

Consider the underactuated rotary crane manipulator system shown in Fig. 4. The first generalized coordinate,θ, is the angular deviation measured from the vertical, of the pendular mechanical load of massmand lengthL. The second generalized coordinate,φ, is the angular displacement of the crane disk, with moment of inertiaJ, The angleφis measured from the vertical, considering its zero value that of the pendular load aligned with the vertical. The pivot is mechanically attached to the crane at a distanceRto the center of the main disk. The system has a control input,τ, acting on the main disk.

The Lagrangian of the system is obtained as

Using the Euler-Lagrange formalism, the full non-linear model of the rotary crane is given by

Fig. 4 Rotary pendubot-like crane

where the linear part ofξ(¨y,y) is measurable thanks to its dependence on the incremental variables,θandφ. However,this term may be considered as an unknown disturbance under a ROESO-ADRC controller scheme. Abusing the notation, by mixing time-domain variables with frequency domain operators, a controller of the form (41) was implemented, via computer simulations, for rest-to-rest trajectory tracking purposes,

4.1 Simulation results

Simulations were performed for the rotary crane system with the following parameters:

The controller performance is also tested, under the influence of a time-variant disturbance signal persistently affecting the system (see Fig. 6). Again the desired output is accurately tracked, and the tracking error tends to zero.However, the angular positions present small oscillations around their desired value, due to the disturbance signal.The control efforts are seen trying to cancel the acting disturbance signal. Notice that in these numerical simulations,the controller overcomes the pointφ=π∕2,θ≈0 , where instantaneously, the controlling effect of the angular acceleration of the disk, acting on the pendulum dynamics, is substantially diminished.

To test the linear controllers robustness, initial conditions were set to be: (φinit,θinit)=(-α,0) , while still using the system linearization around the equilibrium point,(,)=(0,0) . Considering the “lack of controllability” forφaround ±π∕2 , andθ=0 , a trajectory passing through one of these points will evaluate the linear controllers robustness.

Figure 7 shows the system response for a trajectory from(φinit,θinit)=(-2π∕3,0) to (φfinal,θfinal)=(2π∕3,0) , during a time interval of 30 seconds. According to this, independently of the non-linear dynamic of the system, the controller is able to track the reference with an error magnitude of around 3%.

A final test for the controller robustness includes an additive disturbance to the input of the nonlinear system.In Fig. 8, the simulation results demonstrate the controllers robustness exposed to low frequency input disturbances,being able to keep the tracking error around a 3% magnitude.

Fig. 5 ADRC controlled responses of rotary crane

5 Conclusions

Within the context of output reference trajectory tracking tasks, we have shown that the paradigm of flat SISO nonlinear Lagrangian systems, consisting of a perturbed linear chain of integrators, admits a classical controller, in transfer function form, naturally obtained via endogenous injections, using the flat output tracking error and the control input tracking error. The obtained classical controller is the same as the GPI controller, and it can also be made identical, by proper choice of its coefficients, to the reduced order observer based active disturbance rejection controller, written in the frequency domain. This last equivalence is valid provided the separation principle is respected in terms of a stable co-prime factorization of the overall characteristic polynomial in the obtained closed-loop transfer function.These general results are directly applicable in the control of non-feedback linearizable Lagrangian systems which exhibit a controllable (i.e., flat) tangent linearization. The proposed linear output feedback controller is shown to be robust, with respect to neglected nonlinearities and exogenous additive disturbances, when used in the control of the actual nonlinear system. A non differentially flat rotary crane example was presented for which a linear classical controller produces accurate trajectory tracking results in excursions far away from the equilibrium point used for tangent linearization. Successful digital computer simulations are presented.The results are extendable to MIMO nonlinear Lagrangian systems which are either differentially flat (i.e., feedback linearizable via dynamical feedback) or for those systems that while being non-differentially flat, exhibit controllable tangent linearizations around an operating equilibrium point.

Fig. 6 ADRC controlled responses of rotary crane under input disturbance

Fig. 7 ADRC controlled responses of rotary crane trajectory -α to α

Fig. 8 ADRC controlled responses of rotary crane trajectory -α to α under input disturbance

AcknowledgementsThe work of M. A. Aguilar-Orduña and B. C.Gómez-León was supported by Consejo Nacional de Ciencia y Tecnología (CONACYT) Mexico under Scholarship Grants no. 702805 and no. 1039577, respectively.