Fundamental Limits in Combine Harvester Header Height Control

Short abstract

This paper investigates fundamental performance limitations in the control of a combine harvester's header height control system. There are two primary subsystem characteristics that influence the achievable bandwidth by affecting the open loop transfer function. The first subsystem is the mechanical configuration of the combine and header while the second subsystem is the electrohydraulic actuation for the header. The mechanical combine + header subsystem results in an input–output representation that is underactuated and has a noncollocated sensor/actuator pair. The electrohydraulic subsystem introduces a significant time delay. In combination, they each reinforce the effect of the other thereby exacerbating the overall system limitation of the closed loop bandwidth. Experimental results are provided to validate the model and existence of the closed loop bandwidth limitations that stem from specific system design configurations.

1. Introduction

With the world population increasing over the next several decades, agriculture will be called upon to provide greater yields in food production with relatively little increase in land usage. Therefore, it is imperative that efficiencies associated with automation become part of the overall solution. A key aspect is the machinery used to perform the agricultural tasks; one example of this, the combine harvester system, is discussed in this article. The combine harvester is used to extract the crops from the field and during this harvesting process seed loss is a critical problem [1]. It has been estimated that approximately 75% of the crop losses occur at the header [2] and a significant portion of the header loss is caused by improper setting of the header height. Therefore, the header height control problem under study is motivated by the interest in improving the efficiency and productivity of the harvesting process, specifically to increase the harvest yield and decrease the total harvest time.

Figure 1 shows a schematic of a combine harvester system operating in the vertical plane. The header height is defined as the distance between the header tip and ground. By raising or lowering the header with an actuator, usually hydraulic, the header height can be adjusted. If the header height is too large, there is a reduction in harvest yield since much of the viable crop will be left unharvested. Conversely, if the header height is maintained at too low a level, equipment damage or operator fatigue will result.

Fig. 1.

Combine system

The primary solution approach taken to date is the feedback system depicted in Fig. 2. Some look-ahead feedforward approaches have been attempted in industry using laser, ultrasonic, and radar sensors. However, each has been sufficiently challenging as to preclude introduction in practice. The accepted feedback sensor is usually a “feeler” that drags along the ground. Rotation of the feeler relative to the header mount is measured and translated into header height as shown in Fig. 1. The goal is to design the controller $C(s)$ in Fig. 2 so as to regulate the header height to a prescribed setpoint in the face of unknown ground disturbances. Physically, the ground disturbances are induced by the interaction between the ground and the flexible tires; when the combine is driving over irregular terrain, the changing ground height works as disturbance signal to excite the vibration of the combine body. Therefore, the header height control can be seen as the ability to have the header track the changing profile of the terrain at the header tip, and at the same time reject the disturbances under the tires. The higher the closed loop bandwidth (frequency of disturbance that can be rejected or frequency of reference that can be tracked), the more rapid a change in terrain can be accommodated and the faster the vehicle can traverse the field. These all lead to greater efficiency and productivity. Header height control has been a challenging issue in industry for decades, and hence limited harvesting speeds have occurred as a result.

Fig. 2.

Schematic of feedback header height control

While relevant, this control problem has received relatively little attention from the research community. Early approaches of feedback control were proportional-type controllers with an input dead zone operating around the set-point [3]. One of the few recent investigations to utilize modern control techniques introduced a linear quadratic Gaussian controller to automatically track changing terrain shapes [4]. Another reduced order state feedback controller was proposed by using a sky hook damper to simplify an optimal full state feedback controller and reject the output disturbance [5]. The feedback control in Refs. [4] and [5] works well in simulation at relatively low frequencies: below 1 Hz. Field tests illustrate that the achievable bandwidth of a header height control system is usually much lower in practice [6]. However, to increase the working efficiency and obtain desired header height control performance at the same time, a closed loop bandwidth well above 1 Hz is demanded by modern machines. This closed loop bandwidth is desired to accommodate terrain variations resulting from combine forward motion as depicted in Fig. 1. For a desired vehicle speed of approximately 7 miles per hour, which is at the upper limit of current harvesting speeds, the desired closed loop bandwidth specification is 3 Hz or better.

In this article, the authors explore and explain the fundamental causes of the bandwidth limitations in the feedback control of the header height system. The rest of paper is organized as follows. Section 2 introduces the models for the combine system shown in Figs. 1 and 2. The two subsystems that are most relevant to the control limitations are presented: (i) the mechanical subsystem and (ii) the hydraulic actuation subsystem. Section 3 utilizes the models of Sec. 2 and presents an analysis explaining the performance limitation. Section 4 verifies the model and validates the limitation analysis. A conclusion provides a summary and offers insight as to possible remedies that could be undertaken.

2. System Modeling

2.1. Mechanical Subsystem Modeling.

Underactuated systems are those that possess fewer numbers of actuators than the number of degrees of freedom (DOFs). Assume an underactuated manipulator has n independent DOFs, m of which are actuated, and the remaining l = n − m DOFs are termed passive. As illustrated in Ref. [7], the corresponding n generalized coordinates can be written as $q^T=(q_1^T,q_2^T)$, where $q_1\in R^l$ and $q_2\in R^m$ correspond to the passive DOFs and active DOFs, respectively. The dynamic equations of the n DOF system can be written as follows [8]:

$$m_{11}{\stackrel{\mathrm{··}}{q}}_1+m_{12}{\stackrel{\mathrm{··}}{q}}_2+h_1+{\phi }_1=0$$ (1)

$$m_{21}{\stackrel{\mathrm{··}}{q}}_1+m_{22}{\stackrel{\mathrm{··}}{q}}_2+h_2+{\phi }_2=\tau$$ (2)

where the vector functions $h_1(q,\stackrel{·}{q})\in R^l$ and $h_2(q,\stackrel{·}{q})\in R^m$ contain Coriolis and centrifugal terms (likely small in the current application), the vector functions ${\phi }_1(q)\in R^l$ and ${\phi }_2(q)\in R^m$ contain gravitational terms, and $\tau \in R^m$ represents the input generalized force.

The combine system discussed in this paper is such a typical underactuated system, which can be simplified as the planar multibody system shown in Figs. 3 and 4. As such, it contains two rigid bodies: the combine body and the header. There are three DOFS with one actuator amounted between the header and the combine body. The active DOF is the header rotation around the attachment point A with respect to the combine body, and the corresponding generalized coordinate is γ. The two passive DOFs are the combine body rotation and vertical translation relative to its center of gravity, and the corresponding generalized coordinates are θ and $\upsilon$, respectively. The output sensor is installed on the header tip to measure the header height with respect to the ground. Therefore, the sensor is noncollocated with the actuator yet its measurement is influenced by all three DOFs. The mathematical model for this underactuated and noncollocated mechanical system is established as follows.

Fig. 3.

Force analysis for combine body

Fig. 4.

Force analysis for header

Figures 3 and 4 illustrate both the rigid body dynamic analysis and the internally generated forces ($F_{\mathit{Ax}},F_{\mathit{Az}},F_l$) for the combine body and header, respectively. In this combine system, flow control valves are used to lift and lower the header. Assuming the flow compressibility and the cylinder leakage are relatively small, the control input to the mechanical system can reasonably be assumed to be the velocity of the hydraulic cylinder ${\stackrel{·}{l}}_c$. Equations (3)–(12) present geometric relationships between the system variables defined in Figs. 3 and 4. Equations (13)–(15) represent force balances by which the three primary dynamic equations can be represented. Equations (16)–(19) represent relationships among forces, motion of bodies, and external disturbances caused by vertical displacement of the ground. Nomenclature for the variables presented in Eqs. (3)–(19) along with values representative of an actual combine is shown. Exact manufacturer values could not be made available, but the values in Nomenclature are sufficiently accurate to make subsequent analysis valid.

$$l_{\mathrm{c}}^2=l_{\mathrm{ins}}^2+l_{\mathrm{f}}^2-2l_{\mathrm{ins}}{l}_{\mathrm{f}}\cos \gamma$$ (3)

$$l_{\mathrm{ins}}^2=l_{\mathrm{c}}^2+l_{\mathrm{f}}^2-2l_{\mathrm{c}}{l}_{\mathrm{f}}\cos \rho$$ (4)

$$\alpha +\gamma +{\zeta }_h+\beta =\pi /2$$ (5)

$${\zeta }_h+\beta -\rho =\varphi$$ (6)

$$x_{\mathrm{A}}=l_{t1}\cos ({\varphi }_{t1}-\theta )$$ (7)

$$z_{\mathrm{A}}=v-l_{t1}\sin ({\varphi }_{t1}-\theta )$$ (8)

$$x_{\mathrm{cgh}}=x_{\mathrm{A}}+l_{\mathrm{cgh}}\cos (\beta +{\zeta }_h-{\zeta }_{\mathrm{cgh}})$$ (9)

$$z_{\mathrm{cgh}}=z_{\mathrm{A}}-l_{\mathrm{cgh}}\sin (\beta +{\zeta }_h-{\zeta }_{\mathrm{cgh}})$$ (10)

$$\alpha +\theta ={\alpha }_0$$ (11)

$$h=-l_h\sin \beta +z_{\mathrm{A}}+h_0$$ (12)

$$I_h\stackrel{\mathrm{··}}{\beta }=m_{h}g{l}_{\mathrm{cgh}}\cos (\beta +{\zeta }_h-{\zeta }_{\mathrm{cgh}})-F_{\mathrm{l}}{l}_{\mathrm{f}}\sin (\rho )$$ (13)

$$m_{\mathrm{com}}\stackrel{\mathrm{··}}{v}=-F_{\mathrm{A}z}+F_{\mathrm{l}}\sin \varphi -m_{\mathrm{com}}g+F_{\mathrm{f}}+F_{\mathrm{r}}$$ (14)

$$\begin{array}{c}\multicolumn{1}{c}{I_{\mathrm{com}}\stackrel{\mathrm{··}}{\theta }=F_{\mathrm{f}}a-F_{\mathrm{r}}b-F_{\mathrm{l}}{l}_{t2}\sin (-\varphi +{\varphi }_{t2}-\theta )\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+F_{\mathrm{A}x}{l}_{t1}\sin ({\varphi }_{t1}-\theta )-F_{\mathrm{A}z}{l}_{t1}\cos ({\varphi }_{t1}-\theta )}\end{array}$$ (15)

$$F_{\mathrm{A}x}=-m_h{\stackrel{\mathrm{··}}{x}}_{\mathrm{cgh}}+F_{\mathrm{l}}\cos \varphi$$ (16)

$$F_{\mathrm{A}z}=m_h{\stackrel{\mathrm{··}}{y}}_{\mathrm{cgh}}+F_{\mathrm{l}}\sin \varphi +m_{h}g$$ (17)

$$F_{\mathrm{f}}=-k_{\mathrm{f}}(a\theta -z_{\mathrm{f}}+\mathit{v})-b_{\mathrm{f}}(a\stackrel{·}{\theta }-{\stackrel{·}{z}}_{\mathrm{f}}+\stackrel{·}{v})$$ (18)

$$F_{\mathrm{r}}=-k_{\mathrm{r}}(-b\theta -z_{\mathrm{r}}+\mathit{v})-b_{\mathrm{r}}(-b\stackrel{·}{\theta }-{\stackrel{·}{z}}_{\mathrm{r}}+\stackrel{·}{v})$$ (19)

To maintain a desired header height, the absolute header height h in Fig. 4 should track the time-varying ground profile z_h by controlling the cylinder velocity ${\stackrel{·}{l}}_c$. To obtain the open loop transfer function from z_h (the tracking reference) to h (absolute header height), Eqs. (3)–(19) are linearized about an equilibrium point using Eq. (20). The kinematic relationships in Eqs. (3)–(12) are linearized using small angle approximations where appropriate. The equilibrium point considered is a header height of 0.15 m with the vehicle on level ground with corresponding values given in Table 1.

$${\frac{\partial f(\stackrel{\rightharpoonup }{x})}{\partial \stackrel{\rightharpoonup }{x}}|}_{{\stackrel{\rightharpoonup }{x}}_{\mathrm{ss}}}\Delta \stackrel{\rightharpoonup }{x}=0$$ (20)

Table 1.

Variable value at the equilibrium point

Symbol	Value	Symbol	Value	Symbol	Value
α ss	0.124 rad	ρ ss	0.489 rad	γss	1.124 rad
φ ss	−0.167 rad	l c,ss	1.535 m	β ss	0.0227 rad
θ ss	−0.011 rad	v ss	−0.0596 m	x A,ss	2.76 m
z A,ss	−0.948 m	F Ax,ss	113,277 N	F Az,ss	29,881 N
F l,ss	114,878 N	F f,ss	137,267 N	F r,ss	58,733 N
x cgh,ss	4.711 m	z cgh,ss	−1.39 m

where $\stackrel{\rightharpoonup }{x}=\left(\alpha \hspace{1em}\beta \hspace{1em}\theta \hspace{1em}\upsilon \hspace{1em}\rho \hspace{1em}\varphi \hspace{1em}\gamma \hspace{1em}{F}_{\mathrm{l}}\hspace{1em}{F}_{\mathrm{A}x}\hspace{1em}{F}_{\mathrm{A}z}\hspace{1em}{F}_{\mathrm{f}}\hspace{1em}{F}_{\mathrm{r}}\hspace{1em}{x}_{\mathrm{A}}\hspace{1em}{z}_{\mathrm{A}}\hspace{1em}{x}_{\mathrm{cgh}}\hspace{1em}{z}_{\mathrm{cgh}}\right)$, $f(\stackrel{\rightharpoonup }{x})$ represents Eqs. (3)–(19), ${\stackrel{\rightharpoonup }{x}}_{\mathrm{ss}}$ are the steady state value of $\stackrel{\rightharpoonup }{x}$ at the equilibrium point, and $\Delta \stackrel{\rightharpoonup }{x}$ are the deviations of $\stackrel{\rightharpoonup }{x}$ from the equilibrium point.

With the data from Nomenclature and Table 1, we can obtain 17 linear equations from Eqs. (3)–(19). Since the system is a three DOF system, choose the variables $(\Delta \theta \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\Delta \upsilon \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\Delta \gamma )$ as the independent variables, and others in $\Delta \stackrel{\rightharpoonup }{x}$ as dependent variables. The resulting dynamics can be represented by Eqs. (21) and (22), where $q_1={(\Delta \theta \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\Delta \upsilon )}^T$ corresponds to the displacements of the two passive DOFs discussed above, and $q_2=(\Delta \gamma )$ corresponds to the displacement of the active DOF. The coefficient matrices are also given based on the linearized system. The deviation of header height output can be expressed as a linear combination of the displacements of the three DOFs as in Eq. (23)

$$M_1\stackrel{\mathrm{··}}{q}+H_1\stackrel{·}{q}+{\Phi }_{1}q=0$$ (21)

$$M_2\stackrel{\mathrm{··}}{q}+H_2\stackrel{·}{q}+{\Phi }_{2}q=\tau$$ (22)

$$y=\left[\begin{array}{cc}\multicolumn{1}{c}{C_1}& \hfill C_2\hfill \end{array}\right]\left(\begin{array}{c}\multicolumn{1}{c}{q_1}\\ \multicolumn{1}{c}{q_2}\end{array}\right)$$ (23)

where $M_1=\left[\begin{array}{cc}\multicolumn{1}{c}{M_{11}}& \hfill M_{12}\hfill \end{array}\right]$, $M_{11}=m_{11}=\left[\begin{array}{cc}\multicolumn{1}{c}{1}& \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill \end{array}\right]$, $M_{12}=m_{12}=\left[\begin{array}{c}\multicolumn{1}{c}{0.22314}\\ \multicolumn{1}{c}{0.12265}\end{array}\right]$, $H_1=\left[\begin{array}{cc}\multicolumn{1}{c}{H_{11}}& \hfill H_{12}\hfill \end{array}\right]$, $H_{11}=\left[\begin{array}{cc}\multicolumn{1}{c}{1.2764}& \hfill -0.23168\hfill \\ \multicolumn{1}{c}{-1.5577}& \hfill 2.814\hfill \end{array}\right]$, $H_{12}=\left[\begin{array}{c}\multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right]$, ${\Phi }_1=\left[\begin{array}{cc}\multicolumn{1}{c}{{\Phi }_{11}}& \hfill {\Phi }_{12}\hfill \end{array}\right]$, ${\Phi }_{11}=\left[\begin{array}{cc}\multicolumn{1}{c}{77.762}& \hfill -4.0543\hfill \\ \multicolumn{1}{c}{-33.721}& \hfill 155.5\hfill \end{array}\right]$, ${\Phi }_{12}=\left[\begin{array}{c}\multicolumn{1}{c}{-1.0936}\\ \multicolumn{1}{c}{12.539}\end{array}\right]$, $M_2=\left[\begin{array}{cc}\multicolumn{1}{c}{M_{21}}& \hfill M_{22}\hfill \end{array}\right]$, $M_{21}=m_{21}=0$, $M_{22}=m_{22}=1$, $H_2=\left[\begin{array}{cc}\multicolumn{1}{c}{H_{21}}& \hfill H_{22}\hfill \end{array}\right]$, $H_{21}=0$, $H_{22}=0$, ${\Phi }_2=\left[\begin{array}{cc}\multicolumn{1}{c}{{\Phi }_{21}}& \hfill {\Phi }_{22}\hfill \end{array}\right]$, ${\Phi }_{21}=0$, ${\Phi }_{22}=0$, $\tau =k_{\tau }{\stackrel{\mathrm{··}}{l}}_c=1.7016{\stackrel{\mathrm{··}}{l}}_c$, $C_1=\left[\begin{array}{cc}\multicolumn{1}{c}{7.3596}& \hfill 1\hfill \end{array}\right]$, $C_2=4.5989$.

2.2. Hydraulic Subsystem Modeling.

As mentioned in Sec. 2.1, an electrohydraulic actuator is used to control the angle between the header and the combine body. The dynamics in the electrohydraulic system come primarily from the valve. Assuming the fluid is incompressible, the steady state valve flow is proportional to the current command I _in as given in Eq. (24). The flow dynamics are thereby dominated by the second order characteristics between the current command and the actual valve displacement given in Eq. (25)

$$Q_{1,\mathrm{static}}=K\sqrt{\Delta p}(I_{\mathrm{in}})$$ (24)

$$Q_{1,\mathrm{dynamic}}=\frac{Q_{1,\mathrm{\hspace{1em}}\mathrm{static}}}{s^2+2{\zeta }_v{\omega }_{v}s+{\omega }_v^2}$$ (25)

where K is a flow coefficient, $\Delta p$ is the pressure difference across the valve, ${\zeta }_v$ and ${\omega }_v$ are the damping ratio and natural frequency of the valve. For the experimental system, the valve bandwidth was validated as 10 Hz.

If one assumes the pressure difference upstream and downstream of the flow control valve is constant and the valve dynamics are sufficiently high bandwidth, the electrohydraulic system can be further simplified to be a cylinder velocity ${\stackrel{·}{l}}_c$ proportional to the current command I _in with a time delay. As will be seen, the delay incorporates frictional effects [9] in the cylinder seals and linkage bearings. When the system is operating at different header positions, the time delay varies due to the kinematic dependency of the nonlinear friction characteristics in the mechanical system. For clarity of exposition, the delay is assumed to be constant. The hydraulic system dynamics can then be considered as given in following equation, where k _hydr is the corresponding coefficient (see Nomenclature).

$${\stackrel{·}{l}}_c(t)=k_{\mathrm{hydr}}{I}_{\mathrm{in}}(t-T)$$ (26)

3. Fundamental Limitations to Combine Header Height Control

3.1. Underactuated and Noncollocated Systems.

In the following analysis, we will show how the characteristics of the underactuation and noncollocation influence the position of the open loop zeros and poles for the linearized system, and how such zeros and poles would induce limitations on feedback control. First, we examine the mechanical subsystem to illustrate how zero dynamics, or open loop zeros, result from plant dynamics. To make the analysis relevant to the case of the header height problem, only a single input single output system is considered. Defining a coordinate transform matrix $T=\left[\begin{array}{cc}\multicolumn{1}{c}{{\mathit{I}}_{\mathit{m}}}& \hfill 0\hfill \\ \multicolumn{1}{c}{C_1}& \hfill C_2\hfill \end{array}\right]$, the coordinates can be transformed as $\overline{q}=\mathit{Tq}$, where ${\overline{q}}^T=(q_1^T,{\overline{q}}_2^T)$. Since $\left|C_2\right|\ne 0$, the transform matrix T is nonsingular, and this coordinate transformation is valid. The inverse of the matrix T can be calculated as $T^{-1}=\left[\begin{array}{cc}\multicolumn{1}{c}{{\mathit{I}}_{\mathit{m}}}& \hfill 0\hfill \\ \multicolumn{1}{c}{{\overline{C}}_1}& \hfill {\overline{C}}_2\hfill \end{array}\right]$, where ${\overline{C}}_1=-(1/C_2)C_1$ and ${\overline{C}}_2=1/C_2$. Applying the coordinate transformation above to the system in (21) and (22) and substitute generalized force τ with $\tau =k_{\tau }\stackrel{·}{u}$ (u is the output of hydraulic cylinder velocity ${\stackrel{·}{l}}_c$), the new system can be expressed as Eqs. (27)–(29). In the new coordinate system, the output y is not correlated to the passive DOFs q ₁ anymore.

$${\overline{M}}_1\stackrel{\mathrm{··}}{q}+{\overline{H}}_1\stackrel{·}{q}+{\overline{\Phi }}_{1}q=0$$ (27)

$${\overline{M}}_2\stackrel{\mathrm{··}}{q}+{\overline{H}}_2\stackrel{·}{q}+{\overline{\Phi }}_{2}q=k_{\tau }\stackrel{·}{u}$$ (28)

$$y={\overline{q}}_2$$ (29)

where ${\overline{M}}_1=M_1{T}^{-1}$, ${\overline{H}}_1=H_1{T}^{-1}$, ${\overline{\Phi }}_1={\Phi }_1{T}^{-1}$, ${\overline{M}}_2=M_2{T}^{-1}$, ${\overline{H}}_2=H_2{T}^{-1}$, and ${\overline{\Phi }}_2={\Phi }_2{T}^{-1}$. Define a feedback controller as

$$\overline{u}={\tilde{M}}_{22}^{-1}(k_{\tau }\stackrel{·}{u}-{\tilde{H}}_2\stackrel{·}{\overline{q}}-{\tilde{\Phi }}_2\overline{q})$$ (30)

where ${\tilde{M}}_{22}={\overline{M}}_{22}-{\overline{M}}_{21}{\overline{M}}_{11}^{-1}{\overline{M}}_{12}$, ${\tilde{H}}_2={\overline{H}}_2-{\overline{M}}_{21}{\overline{M}}_{11}^{-1}{\overline{H}}_1$, and ${\tilde{\Phi }}_2={\overline{\Phi }}_2-{\overline{M}}_{21}{\overline{M}}_{11}^{-1}{\overline{\Phi }}_1$.

Substituting Eq. (30) into Eqs. (27) and (28) results in the system given in Eqs. (31) and (32). By inspection, the zero dynamics of the system are represented by Eq. (31) since the dynamics of q ₁ do not affect the output y. Note that ${\overline{M}}_{11}$, ${\overline{H}}_{11}$, ${\overline{\Phi }}_{11}$ deviate from the original coefficient matrices of the passive DOFs ($M_{11}$, $H_{11}$, ${\Phi }_{11}$) with terms (${\overline{C}}_1{M}_{12}$, ${\overline{C}}_1{H}_{12}$, ${\overline{C}}_1{\Phi }_{12}$).

$${\overline{M}}_{11}{\stackrel{\mathrm{··}}{q}}_1+{\overline{H}}_{11}{\stackrel{·}{q}}_1+{\overline{\Phi }}_{11}{q}_1={\overline{M}}_{12}{\stackrel{\mathrm{··}}{\overline{q}}}_2-{\overline{H}}_{12}\stackrel{·}{\overline{q}}2+{\overline{\Phi }}_{12}{\overline{q}}_2$$ (31)

$${\stackrel{\mathrm{··}}{\overline{q}}}_2=\overline{u}$$ (32)

where ${\overline{M}}_{11}=M_{11}+{\overline{C}}_1{M}_{12}$, ${\overline{H}}_{11}=H_{11}+{\overline{C}}_1{H}_{12}$, ${\overline{\Phi }}_{11}={\Phi }_{11}+\hspace{1em}{\overline{C}}_1{\Phi }_{12}$, ${\overline{M}}_{12}=M_{12}{\overline{C}}_2$, ${\overline{H}}_{12}=H_{12}{\overline{C}}_2$, and ${\overline{\Phi }}_{12}={\Phi }_{12}{\overline{C}}_2$.

The zero dynamics then can be further represented as Eq. (33). For the underactuated and noncollocated system, expressed as Eqs. (21)–(23), the zeros of the system are the eigenvalues of the matrix N.

$$\begin{array}{cc}\multicolumn{1}{c}{\left(\begin{array}{c}\multicolumn{1}{c}{{\stackrel{\mathrm{··}}{q}}_1}\\ \multicolumn{1}{c}{{\stackrel{·}{q}}_1}\end{array}\right)}& =N\left(\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{q}}_1}\\ \multicolumn{1}{c}{q_1}\end{array}\right)\hfill \\ \multicolumn{1}{c}{N}& =\left[\begin{array}{cc}\multicolumn{1}{c}{-{\overline{M}}_{11}^{-1}{\overline{H}}_{11}}& \hfill -{\overline{M}}_{11}^{-1}{\overline{\Phi }}_{11}\hfill \\ \multicolumn{1}{c}{I}& \hfill 0\hfill \end{array}\right]\hfill \end{array}$$ (33)

The open loop poles also have close and explicit relationship with the dynamics of the DOFs. For an underactuated system, the dynamics of the active DOFs q ₂ are directly determined by the input u without coupling to the passive DOFs q ₁. These coefficient matrices $M_{12}$, $H_{12}$, ${\Phi }_{12}$ are always null. Applying a Laplace transformation on Eq. (22) gives the transfer function from u to q ₂ as

$$q_2(s)={(M_{22}{s}^2+H_{22}s+{\Phi }_{22})}^{-1}{k}_{\tau }\mathit{su}(s)$$ (34)

Additionally, the response of the passive DOFs q ₁ is determined by the behavior of active DOF q ₂. Then, Eq. (21) can be rewritten as

$$M_{11}{\stackrel{\mathrm{··}}{q}}_1+H_{11}{\stackrel{·}{q}}_1+{\Phi }_{11}{q}_1=-M_{12}{\stackrel{\mathrm{··}}{q}}_2-H_{12}{\stackrel{·}{q}}_2+{\Phi }_{12}{q}_2$$ (35)

The transfer function from q ₂ to q ₁ can be obtained by again using Laplace transforms

$$q_1(s)={(M_{11}{s}^2+H_{11}s+{\Phi }_{11})}^{-1}(-M_{12}{s}^2-H_{12}s-{\Phi }_{12})q_2(s)$$ (36)

Substituting Eqs. (34) and (36) into Eq. (23), the transfer function from input u to output y then can be expressed as

$$\begin{array}{c}\multicolumn{1}{c}{y(s)=\frac{C_1(-M_{12}{s}^2-H_{12}s-{\Phi }_{12})+C_2(M_{11}{s}^2+H_{11}s+{\Phi }_{11})}{M_{11}{s}^2+H_{11}s+{\Phi }_{11}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\times \frac{k_{\tau }s}{(M_{22}{s}^2+H_{22}s+{\Phi }_{22})}u(s)}\end{array}$$ (37)

which clearly illustrates how the open loop zeros for the underactuated and noncollocated system are dependent on the passive DOFs and the poles are determined by both the active and the passive DOFs. Particularly in this combine system, the matrices of the active DOF dynamics $H_{22}$, ${\Phi }_{22}$ are null, so all the nontrivial poles are determined by the passive DOFs, which can be calculated by the eigenvalues of the matrix P in Eq. (38). Additionally, the derivative on the numerator cancels one integrator and induces a fifth order system instead of sixth.

$$\begin{array}{cc}\multicolumn{1}{c}{\left(\begin{array}{c}\multicolumn{1}{c}{{\stackrel{\mathrm{··}}{q}}_1}\\ \multicolumn{1}{c}{{\stackrel{·}{q}}_1}\end{array}\right)}& =P\left(\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{q}}_1}\\ \multicolumn{1}{c}{q_1}\end{array}\right)\hfill \\ \multicolumn{1}{c}{P}& =\left[\begin{array}{cc}\multicolumn{1}{c}{-M_{11}^{-1}{H}_{11}}& \hfill -M_{11}^{-1}{\Phi }_{11}\hfill \\ \multicolumn{1}{c}{I}& \hfill 0\hfill \end{array}\right]\hfill \end{array}$$ (38)

The previous analysis can be demonstrated numerically with the linearized combine system. Using the linearized system values from Table 1, and the analysis from (34)–(37), gives

$$P_{\mathrm{com}}(s)=L\left\{\frac{y}{{\stackrel{·}{l}}_c}\right\}=\frac{4.822(s+1.49\hspace{1em}\pm \hspace{1em}12.2i)(s+0.88\hspace{1em}\pm \hspace{1em}11.3i)}{s(s+1.48\hspace{1em}\pm \hspace{1em}12.4i)(s+0.57\hspace{1em}\pm \hspace{1em}8.7i)}$$ (39)

The four nontrivial poles have undamped natural frequencies of 2.1 Hz and 1.4 Hz with corresponding damping ratios of 0.104 and 0.069, respectively. The undamped natural frequencies of the zeros are 2.07 Hz and 1.87 Hz with the damping ratios as 0.114 and 0.0894, respectively. Figure 5 shows the open loop poles and zeros for the system demonstrating an interlacing of poles and zeros close to the jω axis. This type of phenomenon is also present and well recognized in other underactuated control systems, such as flexible structures [10].

Fig. 5.

Open loop poles and zeros of linearized combine mechanical system

It is well known that input/output pairs with low frequency and lightly damped zeros and poles can limit the effectiveness of any feedback control approach [11,12]. Due to the proximity between open loop poles and zeros as shown in Fig. 5, there is difficulty in moving closed loop poles very far from their open loop location. Clearly, this will be a fundamental limitation regardless of any feedback controller C(s).

In the following, a frequency domain analysis of the system limitations is given. Figure 6 gives the closed loop sensitivity function associated with the system of Eq. (39) using a proportional feedback gain tuned by experience to be 0.3. It clearly shows the effects on the magnitude and phase plots caused by the zeros and poles. From the Fig. 6, the closed loop bandwidth is limited to below approximately 1.43 Hz. To obtain a higher closed loop bandwidth, a more sophisticated controller C(s) could be introduced to decrease the sensitivity function over a broader range of low frequencies. However, according to the Bode sensitivity integral (Eq. (40)), any reduction in the sensitivity function at lower frequencies would result in an increase in higher frequency [13].

Fig. 6.

Closed loop sensitivity plot of linearized combine mechanical system

$${\int }_0^{\infty }\ln |s(j\omega )|d\omega =-\frac{\pi }{2}\underset{s\to \infty }{lim}\mathit{sC}(s)P(s)$$ (40)

The effort of this C(s) to improve the sensitivity function in frequency under 1.43 Hz will cause a “piling up” of the sensitivity function at and above 1.43 Hz. This will make the system lose robustness at these higher frequencies possibly leading to instability. Fundamentally, the performance is limited by the position of open loop zeros and poles, which is due to the noncollocated and underactuated nature of this system as shown above. Figure 6 illustrates the challenge faced by any feedback controller in achieving a closed loop bandwidth on the order of the desired 3 Hz value.

This system-level behavior is not unique to the header height control system on a combine. Any underactuated system with a noncollocated sensing and actuation and lightly damped, low natural frequency passive DOFs will introduce similar pole and zero pairs (or even worse, unstable zeros or poles) in the open loop transfer function thereby fundamentally limiting the bandwidth achievable by any controller [14,15]. Below, in Sec. 3.2, we illustrate that the situation is even more challenging when the actuation subsystem contains delays.

3.2. Time Delay Systems.

It is well known that time delays in feedback systems reduce available bandwidth in order to maintain closed loop stability [12,16]. This is true irrespective of the feedback approach taken. Due to the subsystem design, the actuator delay present in the combine header height system is up to 0.3 s as will be illustrated in Sec. 4.1. This time is large relative to the desired closed loop system bandwidth of 3 Hz. The delay can vary somewhat with the hardware configuration but will exist in some form for all header height actuations systems due to cost and manufacturing constraints of these agricultural systems. It cannot be eliminated by means of feedback. According to Ref. [12], the bandwidth limitation caused by a time delay can be expressed by Eq. (41). Therefore, in the present combine system, the time delay from the actuator will limit the closed loop bandwidth to below 0.53 Hz.

$${\omega }_c<\frac{1}{T}$$ (41)

where ω_c is the achievable bandwidth, and T is the delay. This limitation deteriorates the already low bandwidth induced by the mechanical structure characteristics discussed in Sec. 3.1. Figure 7 shows a sensitivity function of the closed loop system from Fig. 6 but including a delay of 0.3 s. Clearly, the situation has deteriorated by the extra phase decrease, thereby making the available bandwidth further below the desired target.

Fig. 7.

Closed loop sensitivity plot of linearized combine system with actuator delay

4. Model Validation

The modeling and analysis of Secs. 2 and 3 posit the existence of fundamental limitations to feedback control for the header height control problem in combines. This section describes the experimental procedures and results of the models created in Sec. 2 to validate the analysis on a real-world system. First, the hydraulic system model given in Sec. 2.1 is validated in the time domain. Subsequently, the mechanical system model is validated in the frequency and time domains. The results of this section indicate the validity of the plant models used in the analysis of Sec. 3.

4.1. Hydraulic Subsystem Model Validation.

To validate the simplified electrohydraulic system from Sec. 2.2, the hydraulic actuation had to be separated from the combine. Figure 8 illustrates a test stand designed to perform this task. The foundation pile acts as reaction wall for the actuator to push on with no pitch or heave dynamics as would be found on the actual combine. A header is also attached to provide a realistic inertial load for the actuator to move. The valve and pump systems are replicated from a production combine system and a height sensor is installed on the header tip to measure the header height. Additionally, several pressure sensors are installed throughout the hydraulic system for diagnostic purposes.

Fig. 8.

Hydraulic system test bed

To measure the time domain response of the hydraulic system, a step command is applied to the control valve and the available signals are monitored as shown in Fig. 9. The sampling rate for the sensors is 20 Hz. There is a combination of responses which include the time to open valves (t1), to stroke the load sensing pump (t2), to build up pressure of the pump output (t3), and to overcome system friction (t4). Summing these effects results in a total time delay T = 0.3 s. This delay constant can vary with different combines and different operation positions. However, for this particular investigation, it is treated as constant for clarity of exposition.

Fig. 9.

Open loop step response of hydraulic system for input current

To verify that a pure time delay (Eq. (26)) represents the major characteristic of the hydraulic system, the closed loop step responses from simulation and experiment are compared in Fig. 10. A proportional controller is used to make the header follow a step reference. The results of Fig. 10 indicate that the simulations results match the response of the actual system sufficiently well so as to have confidence in the simplified model in Eq. (26).

Fig. 10.

Closed loop step response of the test system

4.2. Mechanical Subsystem Model Validation.

The mechanical subsystem experiments were performed on an experimental John Deere combine + header system shown in Fig. 11. For this validation, it was not possible to introduce a perfect actuator and thereby separate the electrohydraulic subsystem from the mechanical subsystem. However, given the validity of the electrohydraulic subsystem demonstrated in Sec. 4.1, the effect of actuation model error affecting the validation of the mechanical subsystem model is minimized. The mechanical subsystem model from Sec. 2 was validated by both frequency domain and time domain responses. Since the open loop system in (24) is type 1, a sine sweep frequency response must be performed in a closed loop manner. For the frequency domain responses, a simple controller was utilized to generate a closed loop transfer function both in simulation and experimentally.

Fig. 11.

Experimental combine used for field test results

A series of sinusoidal height references, constant amplitude with varying frequency, were fed to both simulated and experimental closed loop systems for a stationary combine on level ground. The output heights from the header height sensor were collected and compared with the reference signals. Performing a “frequency by frequency” analysis of magnitude and phase differences allowed for the construction of a frequency response plot in Fig. 12. The result from experiment is compared with the simulation for the nonlinear system given by Eqs. (3)–(19). The nonlinear model fits the physical system well in the magnitude plot. The phase plots have some differences, primarily due to the assumption of a constant friction level, and hence a constant delay, in the hydraulic subsystem. In the experimental system, the friction varies with a change in relative orientation between the combine and the header which is the primary cause of the phase differences. While the nonlinearity of the system would lead to amplitude dependent responses, the nature of Fig. 12 clearly illustrates the matching of the major dynamic modes. By comparing the linearized system with the nonlinear system, the linearization preserves the main model information and can describe the system behavior accurately enough. For further validation in the time domain, see Ref. [6].

Fig. 12.

Closed loop frequency response comparisons between experimental results and simulation

5. Conclusion and Discussion

This paper used system dynamics and common analytical tools to gain insights into the fundamental limits for a combine harvester header height control system. The modeling and analysis of Secs. 2 and 3 clearly demonstrate the challenges present in header height control and the reasons for the relative bottleneck in increasing vehicle speeds. The underactuation and noncollocation properties of the mechanical system determine the position of the open loop poles and zeros, which results in a system bandwidth upper limit. A time-delay limitation from the hydraulic actuator further limits the achievable closed loop performance.

Clearly, a drastic improvement in the overall system performance cannot be achieved solely by feedback control design. To eliminate or decrease the undesirable mechanical characteristics, the mechanical system needs to be redesigned. For example, by replacing the tires with tracks, the DOFs for the system reduce to one and the system becomes fully actuated. Alternatively, active suspensions can be added between the tires and the combine body to increase the number of actuators and therefore eliminate the underactuation in the system. Other possibilities include redesigning key parameters in the system, such as suspension elements, to improve the low natural frequency and lightly damped characteristics of the passive DOFs. As can be seen, there are multiple ways to address the mechanical system problem. However, all of these must be considered in light of realistic cost and design constraints. As for the actuator delay problem arising from the electrohydraulic system, it may be possible to reduce or eliminate the delay with very high performance servo hydraulics. As with the mechanical redesign, these types of system changes would have to be performed under realistic cost and design constraints. High performance servo-actuators may not be appropriate for an all-weather all-terrain agricultural vehicle and may not meet market price points. The work presented here illustrates a practically relevant problem; the search for an optimal solution remains an open control engineering question.

Glossary

Nomenclature

$a,b$ =

the distance in x direction between front/rear wheel axis and gravity center of combine body (2 m;1.3 m)1

$b_{\mathrm{f},}{b}_{\mathrm{r}}$ =

the damping constant of front and rear tires (22,400 kg/s; 26,300 kg/s)

$h_0$ =

the original height of the A point (1.2 m)

${\alpha }_0$ =

the original value of angle α (0.113 m)

$I_{\mathrm{com}},I_h$ =

the inertias of combine body and header with respect to the gravity center and point A separately (66,000 kg m²; 22,000 kg m²)

$l_{t1},l_{t2},l_{\mathrm{cgh}}$ =

structural length (refer to Fig. 3) (2.9 m, 3 m, 2 m, 0.8 m)

$l_{\mathrm{ins}},l_h,l_{\mathrm{f}}$ =

structural length (refer to Fig. 4) (4.6 m, 1.7 m)

$m_{\mathrm{com}},m_h$ =

the masses of the combine body and the header (15,000 kg; 5000 kg)

$k_{\mathrm{f}},k_{\mathrm{r}}$ =

the spring constant of front and rear tires (1,303,720 N/m; 1,673,600 N/m)

$k_{\mathrm{hydr}}$ =

coefficient from valve current to the velocity of the cylinder (0.032 m/s/A)

${\zeta }_{\mathrm{h}},{\zeta }_{\mathrm{cgh}}$ =

structural angle (refer to Fig. 3) (0.3 rad, 0.1 rad)

${\varphi }_{t1},{\varphi }_{t2}$ =

structural angle (refer to Fig. 4) (0.3 rad, 0.5 rad)

$F_{\mathrm{A}x,}{F}_{\mathrm{A}z}$ =

the forces at the point A in x and z directions (variable)

$F_{\mathrm{f},}{F}_{\mathrm{r}}$ =

the forces on the combine body at the front and rear tires (variable)

$F_{\mathrm{l}}$ =

the force from the hydraulic cylinder on the header and combine body (variable)

$h$ =

the absolute header height (variable)

$l_{\mathrm{c}}$ =

the cylinder length (variable)

$x_{\mathrm{A}},z_{\mathrm{A}}$ =

the distances between the combine body gravity center and the point A in x and z directions (variable)

$x_{\mathrm{cgh},}{z}_{\mathrm{cgh}}$ =

the distances between the header gravity center and the point A in x and z directions (variable)

α,ρ,φ =

refer to Figs. 3 and 4 for the geometric meaning of these terms (variable)

$\beta$ =

the header angle with respect to x axis (variable)

$\theta$ =

the pitch displacement of combine body (variable)

$v$ =

the vertical displacement of combine body (variable)

γ =

the angle between header and combine body (variable)