In this paper, the application of an input disturbance observer (IDO)-based control, based on a simple input observer previously proposed and used for engine control, is demonstrated in two case studies. The first case study is longitudinal aircraft control with unmodeled aerodynamic nonlinearities satisfying matching conditions. The second case study is the control of an inverted pendulum on a cart which corroborates the ease of integration of IDO-based control into more complex controllers in situations when the matching condition is not satisfied. Improved robustness is demonstrated on an experimental system including changing the pendulum weight which is shown to have no effect on the overall control performance. In both case studies, if the IDO is not applied, the control performance is poor and leads to unstable operation.

## Introduction

The use of high-gain observers in nonlinear control to compensate for uncertainties, disturbances and modeling errors, has a long history, see the tutorial paper [1] and papers in the special issue [2]. For instance, Khalil and Freidovich [3] use high-gain observer to compensate for the effects of the disturbances and uncertainties and recover the performance of feedback-linearization-based designs in an output feedback-based setting. The related use of observers to deal with uncertainties and alleviate the need for precise model is also pursued in active disturbance rejection control, see Ref. [4] which covers this technique, references, and links with related approaches. The filtered dynamic inversion controller is shown to provide command following and disturbance rejection for minimum phase uncertain time-invariant systems affected by unknown time-dependent disturbances in Ref. [5]. Liu and Peng [6] study the integration of a disturbance observer, based on an assumption of an additive time-dependent disturbance, into control systems for robot manipulators and provide stability analysis results for the case when the disturbance is constant. The book by Chen and Patton [7] considers the use of input disturbance observer (IDO) for fault detection. From the practical perspective, the IDO-based control has the potential to reduce the time and effort necessary for model-based controller design and calibration by reducing the need to use accurate models [8] as the uncertainty and model errors are compensated by IDO. Main implementation requirements for successful application of IDO-based control involve accurate measurement of system states and outputs, sufficiently high sampling frequency, sufficient actuator authority, and suitable structural properties of the system such as minimum phase characteristics.

In Refs. [912], a simple low-order input observer has been exploited and experimentally validated for several applications in the automotive engine domain. The main contribution of this paper is showing that this IDO can be integrated into practical control schemes and demonstrates the benefits of IDO-based control through two case studies. The first case study concerns the control of the aircraft longitudinal dynamics with aerodynamic uncertainties. This system has been previously treated in Ref. [13] using adaptive control techniques. We demonstrate that effective controller can also be obtained by exploiting simple IDO-based control techniques. In this case study, the uncertainty satisfies the matching conditions in Ref. [14], and semiglobal asymptotic stability properties are guaranteed by theoretical results from Ref. [14] that are briefly reviewed in this paper.2 The second case study involves an experimental system which is an inverted pendulum on the cart. This system represents a benchmark in control theory with dynamics similar to practical systems such as reusable rockets and launch vehicles [15,16]. Even though the matching conditions are not satisfied for this system, as we demonstrate, a cascaded control design can be pursued with IDO-based controller integrated in the outer loop. The closed-loop stability analysis is performed by checking eigenvalues, extensive simulations, and experiments.

The experimental validation of IDO-based control as a part of an integrated cascade control design is an important contribution as due to high-gain characteristics of the IDO it is unclear if such an approach can be successful in practical applications where signals are noisy and dynamics are complex, nonlinear, and potentially nonminimum phase. We believe that the demonstrated robustness and performance recovery coupled with the simplicity of the IDO design and small computational footprint support the perspective that this approach can be attractive to practitioners. The simplicity and ease of understanding of a control design is particularly important in it being adopted for industrial use.

Furthermore, our successful experimental validation of an IDO in a cascade design for an inverted pendulum on the cart is of interest in view of other potentially related techniques, such as backstepping through dynamic surface control [17], as it provides motivation for further research into their practical use in experimental applications.

## Disturbance Estimation and Compensation Based on a Simple Observer

Following Ref. [14], the IDO-based application is easiest for systems with matched disturbances/uncertainties of the form
$x˙=Ax+B(u+w)$
(1)
where x is the state vector, u is the control vector, and w denotes the unmeasured disturbance or uncertainty matched to the control input. The disturbance/uncertainty can be a function of both the state x and time t, i.e.,
$w=η(x,t)$
(2)
Suppose a full state feedback controller for Eq. (1) with w = 0 has been designed that has the following form:
$u=ud+K(x−xd)$
(3)
where ud is a constant feedforward, and xd is the corresponding steady-state equilibrium satisfying Axd + Bud = 0. The controller is assumed to be stabilizing, i.e., the matrix
$Acl=A+BK$
is Hurwitz (asymptotically stable). The Lyapunov function for the closed-loop system with w = 0 can be constructed as
$V1=12(x−xd)TP(x−xd)$
(4)
where P = PT > 0 is the solution to the Lyapunov equation
(5)
For the system model (1) with w ≠ 0, we employ the control law (3) with the compensation of the disturbance
$u=ud+K(x−xd)−ŵ$
(6)

where $ŵ$ is an estimate of w.

To generate $ŵ$, we define an output, z, with dimension equal or greater to that of u
$z=Cx$
(7)
so that the matrix CB that represents the first Markov parameter has a left inverse (CB)L, where $XL=(XTX)−1XT$. Then
$z˙=CAx+CB(u+w)$
(8)
The system (8) is in the form to which a simple observer described in Ref. [9] is applicable. This observer is described as follows:
$ζ˙=γ(r̂+CAx+CBu)r̂=γz−ζŵ=(CB)Lr̂$
(9)
Defining the estimation error
$w̃=w−ŵ$
(10)
and
$V2=12w̃Tw̃=12(w−ŵ)T(w−ŵ)$
(11)
we can consider the following Lyapunov function candidate for the integrated closed-loop system:
$V=V1+V2$
(12)

where V1 is defined in Eq. (4), and V2 is defined in Eq. (11). The following results hold:

Proposition 1 (see Ref. [14]). Suppose$w(t)=η(x(t),t)$, where η is unknown but satisfies the time rate of change quadratic bound, i.e.,
$w˙Tw˙≤ν+x̃TQ1x̃+w̃TQ2w̃ if ϵ2≥V(t)≥ϵ1$
(13)

for some$ν≥0, Q1=Q1T≥0, Q2=Q2T≥0, ϵ1>0$,and ϵ2 > ϵ1. Then, there exists γ*≥ 0 such that for all γ > γ*, ϵ2 ≥ V(t) ≥ ϵ1 implies that$V˙(t)≤0$and V(t) → ϵ1[0, 1] as t → ∞.

Proposition 2 (see Ref. [14]). Suppose Eq.(13)holds globally with ϵ1 = 0 and ϵ2 = +∞. Then, for any ϵ*> 0, there exists γ*≥ 0 such that for all γ > γ*, V(t) ≥ ϵ* implies that$V˙(t)≤0$and V(t) → ϵ*[0, 1] as t → ∞. Thus, the origin of the closed-loop system is semiglobally practically stable.

Remark 1. Under the assumptions of Proposition 2, the desired equilibrium is practically semiglobally stable that is by increasing γ > 0 an arbitrary small neighborhood of the origin can be made attractive from an arbitrary large domain of initial conditions. There is no requirement for $w˙(t)$ to converge to zero as t for these results to hold. For further discussion of semiglobal practical stability, see, e.g., Ref. [18].

Remark 2. Note that for our linear model and w = η(x), it follows that
$w˙=∂η∂x((A+BK)x̃+Bw̃)$

Assuming that $∂η/∂x$ is bounded on compact sets, the bound on $w˙Tw˙$ in Proposition 2 can be made to hold with ϵ1 = 0 and ϵ2 > 0 arbitrary large by choosing suitable Q1 > 0 and Q2 > 0 and ν = 0. This implies that the equilibrium defined by $x̃=0, w̃=0$ is semiglobally stable. This is a stronger result than semiglobal practical stability but it holds since w is not a function of time. If w = η(x, t) were a function of time, but the derivative $∂η/∂t$ were bounded, then the result in Proposition 2 would give only semiglobal practical stability.

## Aircraft Control—Simulation Study

To illustrate the application of Eqs. (6) and (9), we consider a challenging nonlinear model [13] that reflects short-period dynamics for a generic transport aircraft at 0.8 Ma and 30,000 ft with aerodynamic moment uncertainties
$[α˙θ˙q˙]=[−0.701800.9761001−2.69230−0.7322][αθq]+[−0.05730−3.5352](δe+w)$
(14)
Here, α is the angle of attack, θ is the pitch angle, q is the pitch rate, δe is the elevator control signal, and $w=η(x)=0.1 cos(α3)−0.2 sin(10α)−0.05e−α2$ is the unknown nonlinearity. This system has been treated in Ref. [13] using adaptive control techniques. Here, we demonstrate that it can be effectively controlled using a simpler IDO-based design, with the elevator controlled as
$δe=u=−K(x−xd)−ŵ$
(15)

where $ŵ$ is an estimate of w, and xd is the set-point.

The feedback gain K in Eq. (15) has been designed to provide good response without uncertainties as K = [0.7616,−0.6365, −0.5142]. The observer gain has been set to γ = 300.

The aircraft longitudinal responses with compensation off and on are presented in Figs. 1 and 2. The benefit of disturbance canceling is clearly demonstrated with good tracking performance, while the control with no compensation is poor. Since in this example, the disturbance satisfies matching assumptions, and $∂η/∂x$ is bounded on compact sets, the desired equilibrium xd is semiglobally asymptotically stable that its domain of attraction can be made arbitrary large by increasing the gain γ. We have further verified by simulation (plots not shown) that with the unknown time-dependent nonlinearity, in the form $w=η(x,t)=sin(t)[0.1 cos(α3)−0.2 sin(10α)−0.05e−α2]$, our IDO-based control strategy leads to stable operation (with semiglobal practical stability properties, see Remark 2). The transient response of θ and α is similar to the ones in Fig. 1, of course the control and disturbance estimate are different versus the ones in Fig. 2.

Fig. 1
Fig. 1
Close modal
Fig. 2
Fig. 2
Close modal

## Inverted Pendulum Control: Real-Time Control Application

A laboratory inverted pendulum on a cart is an unstable, nonlinear, underactuated system with the nonminimum-phase behavior. The complex dynamics of inverted pendulum on a cart has been a motivation to design and experimentally demonstrate many control algorithms. One of the traditional approaches to control of this system is to linearize the mathematical model at the upright position and based on such a linearized model to design a controller. The problem becomes more challenging if the nonlocal behavior of the closed-loop system is to be explored, especially with uncertainties. In Sec. 4.1, we discuss the design of an IDO-based controller and experimentally verify and validate it on the inverted pendulum test bench, shown in Fig. 3. The pendulum is mounted in a free rotary joint that is fixed to the cart that can move in the horizontal direction. The cart can slide on the rail, and it is driven by the stepper motor via toothed belt. The position of the cart and the angle of the pendulum are measured with the incremental encoders.

Fig. 3
Fig. 3
Close modal

### Mathematical Model and Input Disturbance Observer-Based Control Design.

The inverted pendulum on a cart, schematically shown in Fig. 4, can be described by the following equations:

Fig. 4
Fig. 4
Close modal
$(M+m)q¨+mlθ¨ cos θ−mlθ˙2 sin θ=F$
(16)
$lθ¨+q¨ cos θ−g sin θ=−bθθ˙$
(17)

where M is the mass of cart, m is the mass of pendulum, q is the position of cart ($q˙$ velocity, $q¨$ acceleration), θ is the pendulum angle ($θ˙$ angular velocity, $θ¨$ angular acceleration), g is the acceleration due to gravity, bθ is the damping coefficient for pendulum, and F is the external force acting on the cart.

The stepper motor can operate at the corresponding speed value if the speed set-point is on or underneath the pull-in curve [19]. For any load torque value in this region, the motor can start, stop, or reverse “instantly” (at the corresponding speed set-point value). Therefore, the reaction of the pendulum arm on the cart can be neglected, and the acceleration of the cart $q¨=u$ is chosen as the input.3 In other words, the stepper motor represents an ideally stiff torque source, meaning the external forces acting on the cart will have no effect on the overall cart acceleration. This includes the reaction forces of the pendulum $+mlθ¨ cos θ$ and $−mlθ˙2 sin θ$ acting on the cart, Eq. (16). It is therefore reasonable to neglect these forces even though they are present. By introducing the state vector $sT=[q,q˙,θ,θ˙]$, the equations of motion have the form4
$s˙1=s2s˙2=us˙3=s4s˙4=−bθls4−ulcos s3+glsin s3$
(18)
The model is written in the form $s˙=Fs+Gu+Rw$
$[s˙1s˙2s˙3s˙4]=[010000000001000−bθl][s1s2s3s4]+[010− cos s3l]u+[000−1l]w$
(19)
where $w=−g sin s3$. Note that the disturbance w is not matched to the input u. Since the first two equations are coupled with the last two equations only through the input u, we base the design of disturbance rejection on the pendulum dynamics only (states ) for which the model has the form of Eq. (1) with matched disturbance
$[x˙1x˙2]=[010−bθl][x1x2]+[0−1l](ũ+w(x,t))$
(20)

where $x1=s3, x2=s4, ũ=u cos x1$, and $w=−g sin x1$. This treatment is relevant to similar systems such as launch vehicles for which an inverted pendulum on a cart serves as a prototype [15,16]. The disturbance also accounts for uncertainties and inaccuracies in other model coefficients, including the damping coefficient bθ.

Note that the system (20) is in the form satisfying the matching condition to which IDO-based controller can be directly applied. Note that if the pendulum angle is controlled to a set-point, x1,r, in steady-state, $u=g tan(x1,r)$, suggesting that x1,r behaves similarly to the cart acceleration. Hence, the positioning of the cart is indirectly controlled through the pendulum reference angle x1,r where
$x1,r=Kr[(s1−s1,r)s2]T$
(21)

and where Kr = [kr,1, kr,1] is the reference angle control gain with s1,r being the cart's reference position.

The integrated cascade control law with the integral action added for zero offset tracking of the cart position set-point has the following form:
$u=−Kiei−1 cos x1[K[(x1−x1,r)x2]T−ŵ]$
(22)
where Ki is the integral control gain, K = [k1, k1] is the pendulum control gain, $ŵ$ is the disturbance estimate computed with Eq. (9) (where C = [0, 1]), and ei is the integrated error, a solution to
$e˙i=s1−s1,r$
(23)

This control scheme for the inverted pendulum system is shown in Fig. 5. In addition, a state observer, the extended Kalman filter, Table 6.1-1 in Ref. [20], is applied to the inverted pendulum system to estimate the speed signals of cart and pendulum. For the implementation purposes, the scheme is further decretized with the forward-Euler method with the integration step interval h. The control action and measurements are updated with sampling interval T. Numerical settings of parameters are summarized in Table 1. The identification procedure of the friction parameter value is described in the Appendix.

Fig. 5
Fig. 5
Close modal
Table 1

Numerical settings of parameters for nominal system

 Sampling interval T 0.01 s Integration step h 0.001 s Control gain K [−22.2025, −4.9445] Control gain of reference angle Kr 0.2*[−0.9, −1] Integral control gain Ki −1.5 IDO gain γ 50 Friction parameter bθ 0.0966 m s−1 Center of gravity distance l 0.35 m
 Sampling interval T 0.01 s Integration step h 0.001 s Control gain K [−22.2025, −4.9445] Control gain of reference angle Kr 0.2*[−0.9, −1] Integral control gain Ki −1.5 IDO gain γ 50 Friction parameter bθ 0.0966 m s−1 Center of gravity distance l 0.35 m

Note that Eq. (22) contains singularity if $cos x1=0$. This singularity is away from the desired equilibrium (i.e., it does not affect the local stability) and reflects the physics of the problem that horizontal cart acceleration creates no moment on the pendulum if the pendulum is momentarily in a horizontal position. In the experimental implementation, saturation constraints were imposed on the control signal consistent with the control magnitude limits.

### Stability.

Local asymptotic stability of the closed-loop system at the open-loop unstable equilibrium is verified through eigenvalue analysis. The system equations are linearized ($cos s3=1, sin s3=s3$), and with respect to Eqs. (9), (19), and (23), the closed-loop system dynamics matrix $A¯cl$ has the form
$A¯cl=[010000000000000100000−bθl00000γ2−γbθl−γ0100000]+[010−1/l−γ/l0]·(−[00k1k20Ki]T·(I−[000000000000kr,1kr,20000000000000000000000])−[000γ(CB)L−(CB)L0]T)+[000g/l00]·[001000]T$

where I is the identity matrix. The eigenvalues of $A¯cl$ matrix with altering IDO's gain γ are plotted in Fig. 6 using the parameters displayed in Table 1. It can be seen that the proposed control scheme stabilizes the system in the unstable equilibrium for any positive observer gain γ, except the case when the IDO is off, γ = 0. From the practical point of view, γ = 50 was found to give a good performance.

Fig. 6
Fig. 6
Close modal

We note that the linearization-based asymptotic stability analysis is only valid locally. Since the inverted pendulum on a cart does not satisfy our conditions, e.g., the matching assumption, of our semiglobal stability results reviewed in Sec. 2, we leave the analytical study of the domain of attraction of the closed-loop system to future work. In a practical sense, closed-loop simulations and experiments demonstrate large domain of attraction of the desired equilibrium and more importantly a marked improvement in robustness through the addition of the IDO-based compensation.

### Experimental Results.

Simultaneous stabilization of the inverted pendulum and cart position is a challenging problem where the main disturbance to the control system comes from the tipping (accelerating) pendulum due to the gravity force. Careful tuning of the competing cart positioning control loop and pendulum stabilization control loop needs to be accomplished in order to achieve good performance. The performance of proposed control scheme is tested in the following scenario: The pendulum is initialized in the inverted position, and the cart is initially positioned. After the initialization, first 0.5 m magnitude square-wave cycle of the reference cart position is induced. Through the control experiment at the beginning of second 0.5 m magnitude square-wave cycle, the IDO compensation is turned off in order to demonstrate the influence of IDO. Approximately in the middle of the second cycle, the IDO compensation is turned on back again. The IDO is turned off by setting the estimated disturbance signal $ŵ$ in the control law, Eq. (22), to zero.

#### Nominal System.

The first experiment is performed with the original pendulum weight—the nominal weight. The position and speed of the cart are shown in Fig. 7, and the position and speed of the pendulum are shown in Fig. 8. The oscillations between 41 and 47 s are associated with the turned off IDO where we can observe that the system is becoming unstable. At 47 s, the IDO is turned on again and that immediately attenuates the oscillations. The estimation of disturbance is plotted in Fig. 9. The control signal computed by the control law is shown in Fig. 10, and the integrated control signal converted to the angular speed set-point signal is shown in Fig. 11.

Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal

#### Perturbed System.

In the second experiment, an unknown weight is attached to the pendulum. The attached weight changes the center of gravity and natural frequency. Prior to the attachment of weight, no modifications are done on any of the control gains or the IDO gain. The same control scenario as for the nominal system is performed. The position and speed of the cart are shown in Fig. 12, and the position and speed of the pendulum are shown in Fig. 13. The oscillations between 41 and 45 s are associated with the turned off IDO where the system has sufficient damping and eventually would reach the equilibrium. At 45 s, the IDO is turned on again and that immediately attenuates the oscillations. The disturbance estimate is plotted in Fig. 14. The control signal computed by the control law is shown in Fig. 15, and the integrated control signal converted to the angular speed set-point signal is shown in Fig. 16. To confirm the local stability, eigenvalue analysis was performed for the model with varying center of gravity distance parameter l (that reflects the mass variation) for the values between $l∈〈0.05;0.5〉 m$. All the eigenvalues of $A¯cl$ matrix with IDO's gain γ = 50 are in the open-left half plane, and hence the closed-loop system is asymptotically stable.

Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal
Fig. 15
Fig. 15
Close modal
Fig. 16
Fig. 16
Close modal

Both nominal system and perturbed system have been also controlled with the linear quadratic controller based on the linear system model coming from the linearization of Eqs. (16) and (17). The comparison (not shown in the paper) revealed substantial advantage of the IDO-based control compared to the linear quadratic controller in terms of cart positioning dynamics with faster maneuvering ability, better robustness, and larger region of attraction.

## Conclusion

In this paper, we presented two case studies involving a simple IDO-based control scheme [14]. In the first case study, unmodeled aerodynamic nonlinearities have been compensated without resorting to arguably more complex adaptive control scheme as in Ref. [13]. In the second case study, IDO-based control has been integrated into a larger cascade controller for the inverted pendulum on the cart. Experiments conducted on the actual system demonstrated robustness to unmodeled dynamics and to the additional unknown weight arbitrarily positioned on the pendulum.

As these case studies demonstrate, the IDO-based control can be effective in improving the robustness of the nominal control loop and reducing the control design and calibration effort. As confirmed through the actual experiments on the inverted pendulum on the cart system, the high-gain characteristics of the IDO-based control do not present obstacles to real-time implementation. A careful comparison with other control approaches for our case study and the accommodation of less restrictive conditions on the uncertainty is beyond the page limit of this brief paper and is left to future work.

## Acknowledgment

The first and third authors have been supported by the VEGA 1/0301/17 project.

## Funding Data

• Agentura na Podporu Vyskumu a Vyvoja (Project Nos. APVV 14-0399 and APVV-0015-12).

• National Science Foundation (Award No. EECS 1404814).

### Appendix: System Identification and Model Validation

Offline system identification procedure is performed to estimate the model parameter of inverted pendulum system. Two main dynamic subsystems form the inverted pendulum system:

1. (1)
Subsystem is the pendulum dynamics, Eq. (20). In this model, the friction parameter bθ is unknown and needs to be estimated. The acceleration impulse is applied to the motor to excite the pendulum dynamics. The time domain data are processed with the spafdr matlab function, and the pendulum model, Eq. (20), is fitted to the frequency characteristic shown in Fig. 17
Fig. 17

Frequency response of cart acceleration to pendulum position

Fig. 17

Frequency response of cart acceleration to pendulum position

Close modal
. The matlab ssest function is used to estimate the friction parameter bθ [21].
2. (2)
Subsystem consists of cart dynamics driven by the stepper motor directly connected to the pulley–teeth–belt mechanism that drives the linear motion of cart. The aim of the following experiment is to directly validate the dynamic response of cart's position to commanded angular speed set-point of the stepper motor. The relationship of angular speed set-point of the stepper motor and the position of cart is given by the first-order integrator equation
$s˙1=s2 (m s−1)=2πrp60ui$
(A1)
where ui (rpm) is the angular speed set-point of the stepper motor, and rp = 0.035 m is the radius of pulley. Since the radius is a known parameter, the model given by Eq. (A1) is directly validated against the measured position dynamics of the cart. To excite the position of cart around its centered position, chirp signal (sweeping frequency) experiment is designed to modulate the angular speed set-point with the sinusoidal signal. The measured frequency characteristic for different angular speed set-point amplitudes is shown in Fig. 18 where it is verified that the angular speed set-point of the stepper motor changes the position of cart according to Eq. (A1). For the frequency span 0.01–0.1 Hz, this model is not accurate due to the frictionlike phenomena known as the eddy currents [19]. It is also not accurate above the input frequency 2 Hz where the stepper motor starts to loose the synchronicity accompanied with dithering phenomena. The matlab spafdr function is used to estimate the frequency characteristic [21].
Fig. 18
Fig. 18
Close modal
2

Reference [9] discusses integration of simple input observer into control as well, but the result it contains is different from Ref. [14] and application it addresses is different from the present paper or Ref. [14]. The inverted pendulum system, further considered as a case study in this article, represents a more complex system than the system discussed in Ref. [9].

3

The input u (acceleration) needs to be integrated with respect to time in order to obtain the reference cart velocity converted further to the angular speed set-point ui which serves as the input for the stepper motor.

4

From the engineering perspective, this is a valid simplification of the inverted pendulum on cart equations (Eqs. (16) and (17)) for the driving stiff force (torque) and to the authors' knowledge not previously reported in the literature. It is also validated by our experiments.

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