Controls and experiments: Lessons learned - Department of ...

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Presenting four basic lessons derived from combining control theory and experimental implementation.

By Andrew Alleyne, Sean Brennan, Bryan Rasmussen, Rong Zhang, and Yisheng Zhang

T

his article describes different lessons that can be learned by including experimental aspects in control system research. Several key lessons are identified, and then each lesson is developed within the context of a particular experimental system. A variety of physical experimental systems are used to illustrate that the key lessons need not all be found in the same system, but should one be working with a variety of systems, it is likely that one or more of these issues would arise. The actual

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experimental systems incorporate the fields of vehicle dynamics, air conditioning and refrigeration, and fluid power. However, it is felt that the main points of this article can easily be applied to many other fields.

Background From a historical perspective, experimental aspects of control systems significantly predate their mathematical analysis [l]. Between the 1950s and the 1980s, however, a great deal of emphasis was placed within academia on the

0272-1708/03/$17.00©2003IEEE IEEE Control Systems Magazine

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analytical aspects of feedback control. These included the valuable concepts we now take for granted such as controllability, observability, stability, optimality, realizations, and robustness [2]. It is only within the past two decades that the research pendulum has swung, and we have seen an increased interest in experimental and application aspects of controls. Examples of this are the creation of journals such as IEEE Transactions on Control System Technology, IFAC Control Engineering Practice, and IEEE/ASME Transactions on Mechatronics. Additionally, conferences aimed at implementation and experimental aspects of control have recently developed, including the IEEE Conference on Control Applications, the IEEE/ASME Conference on Advanced Intelligent Mechatronics, and the IFAC Conference on Mechatronic Systems. The focus of this article is on the role of physical experiments in control system research, particularly within academia, and how experiments and theoretical analysis can be performed in a synergistic fashion. Prior to detailing particular experiments and insights gained, we differentiate between two types of approaches to experiments. We believe there is a difference between what we define as control validation experiments and control technology experiments. Although this may seem to be semantic, we clarify the distinction in the discussion. The reader should know that we are not advocating one versus the other, but rather we are merely making an observation. As will be seen in later sections, we believe both are valuable. Control validation experiments are physical systems constructed with the purpose of testing or demonstrating a particular control technique or set of techniques. Care is often taken in the design of the hardware and software so that the key relevant attributes, and only those key attributes, are present in the final experimental system. ln this sense, the experiment that supports the research of control analysis and synthesis is effectively secondary. Aspects of the experiment violating the control researcher’s assumptions are removed by design iterations. An example of this is to size the actuation system so that saturation is not an issue, unless the control methodology being investigated explicitly provides saturation compensation. A specific physical example of a control validation experiment is wheeled mobile robots [3], [4] that utilize dead reckoning and encoder readings of the wheels to determine position. Several of these systems design the wheels to minimize slip, and they operate at low speed so as to maintain a kinematic representation of the system. Therefore, the physical systems retain a level of fidelity to the analytical models that several path-planning and control algorithms utilize. Other specific physical examples are inverted pendulum systems [5], [6], where the uncertainty is small and the model form is well known. The motivation for control validation experiments includes the demonstration and validation of a theoretical

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concept. It is often valuable to show that the theoretical construct can have application to physical systems. However, one problem with complete justification is that the physical systems used for demonstration are usually designed to work well with the theory. Therefore, the theory may not transfer as well to plants dissimilar to the demonstration system. Control validation experiments can also be valuable for determining the practical limitations of a particular algorithm and can point the researcher in new directions to further refine their techniques. Finally, there is usually some level of excitement associated with demonstrating a physical device to others in the field. Control technology experiments occur when a physical system, designed and built a priori, needs to be controlled and is presented to the researcher warts and all. Such a case is representative of industrial practice, and a great deal of effort must go into understanding the physical system and determining the best control approach. Often the fundamental physical process cannot be changed, and undesirable plant characteristics have to be accounted for. A successful example of this is the control of internal combustion engines. Delays, combustion kinetics, and time-varying system dynamics led researchers to consider key advances such as controlling in the crank angle domain [7], [8] to achieve specific control metrics. The primary motivation for conducting control technology experiments is to determine the benefits that embedded intelligence has on engineered products and services. Reducing vehicle emissions by orders of magnitude, as a result of tight air-fuel ratio control, has had a tremendous societal impact [9]. The same is true for the servo controls that allow data storage densities to achieve their current high levels [10]. Additionally, in the process of developing controllers for engineered systems, it is possible to develop fundamental understanding of key dynamical phenomena that can be useful well beyond the field of control. Finally, by examining the control of actual engineered systems, it is possible to identify key factors that limit control effectiveness such as sensor locations and actuation authority. These factors can then be used in system redesign, as was the case when disk drives changed from linear to rotary voice coil motors in the late 1960s [10].

The Lessons Whether the reader is interested in control validation experiments or control technology experiments, we present four basic lessons learned from combining control theory and experimental implementation. There are clearly more points that could be made, and although the following are open to a healthy debate, we choose to focus here on a particular subset due to space constraints. These key lessons include a subset of the most relevant ones developed during the 1999 NSF Workshop on the Integration of Modeling and Control for Automotive Sys-

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tems [11]. Although the lessons learned originated from automotive applications, we believe they hold true for many controlled physical systems. ● Lesson 1: Where possible, modeling should be done within a control-oriented framework. A rule of thumb developed during the aforementioned workshop was that, within this framework, models must be based on straightforward physics and capture approximately 80% of the relevant dynamic behavior. ● Lesson 2: To avoid a control design quagmire, it is

systems-level redesign at a reasonable cost. In Lesson 2, the notion of a well-posed control objective with a priori specifications may seem obvious until examined closer. Often, difficult control problems may be the result of an unfortunate problem formulation. Moreover, it is important to understand what constitutes the minimum acceptable performance for a given task. Using a semiconductor wafer stage example, the cost associated with reducing the maximum absolute value of the scanning error from 50 to 5 nm can be millions of dollars. Each increase or relaxation of performance specifications has quantitative effects on the cost and achievable performance of any overall control system above and beyond topics such as asymptotic versus exponential stability. Lesson 2 becomes particularly important when dealing with control technology experiments. Related to specifications is the notion in Lesson 3 of choices associated with actuation and sensing. Cost or physical constraints may limit the number of sensors and actuators, their placement within a system, and the finite actuator power or sensor resolution. As a general example, the actuators and sensors should be roughly colocated for better performance and easier design [12]. Lesson 4 indicates that when doing experiments, one should always keep an open mind to identifying fruitful avenues of discovery. Many of these avenues are revealed only after confronting what were initially perceived roadblocks for the original investigations. We will examine these four lessons learned within the context of several experimental systems developed by our research group at University of Illinois, Urbana-Champaign (UIUC) [13]. These experiments include both control validation experiments and control technology experiments.

Control validation experiments are physical systems constructed with the purpose of testing or demonstrating a particular control technique or set of techniques. important to know what the appropriate goal is and when the system performance is sufficient to meet specifications developed prior to the control design. ● Lesson 3: The effects of actuation authority and measurement sensitivity choices on achievable closedloop performance should be well understood. ● Lesson 4: The act of serendipity associated with discovery is healthy for research. At first glance, these four points may sound obvious and vacuous to the reader, but there is an importance and subtlety to each. The control-oriented framework of Lesson 1 runs counter to many fields of modeling where effort is given toward increasing a model’s accuracy. At some point, however, the model should be good enough for a model-based controller to have a high probability of success in achieving specified performance goals. The potentially controversial quantitative evaluation of modeling sufficiency is the result of a careful thought process about knowing when a model is good enough. This subtle point is the reason why vehicle cruise control works well but engine cold start emissions control does not [11]. In addition to knowing that a model has captured most of the dynamics, it is important to know where the rest of the uncertainty is coming from. Certainly, there are robust control tools that can provide some level of guarantees with respect to an uncertainty representation for given types of systems. For the practicing control engineer, however, it is sometimes more important to know what causes that uncertainty, not just that it exists. The knowledge is greatly aided by the physics-based representation mentioned above. Should it be difficult to design a controller to meet performance specifications, the control engineer can work with structural, design, and power engineers to see where the dynamic uncertainty can be reduced via overall

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Importance of Control-Oriented Modeling The experimental system examined here is a vapor compression cycle, more commonly known as an air-conditioning or refrigeration cycle. We focus on a transcritical CO2 cycle and its related components, although most commercial systems use a subcritical cycle. These systems are complex devices due to their nonlinear thermo-fluid behavior, and significant savings in energy can be achieved if they are properly controlled. Moreover, with the use of alternative refrigerants such as CO2 , it is possible to reduce a key contributor to ozone layer depletion. The CO2 cycle is a control technology experimental system because the basic thermo dynamic cycles of the air-conditioning system have been well defined for many decades [14], [15] and the control designer may be constrained by the already established process characteristics.


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A typical transcritical air-conditioning system consists of the five components shown in Figure 1. For a more complete description of the working principles of this system, see [16]. Unique aspects of this system versus subcritical ones include the internal heat exchanger that increases the inherently coupled nature of the dynamics and the supercritical state of the refrigerant in the gas cooler. The four controllable inputs to the system are compressor speed, expansion valve opening, and the mass flow rates of air across the evaporator and gas cooler. The outputs of interest are the superheat temperature (a measure of efficiency), evaporator outlet air temperature (a measure of comfort), and the operating pressures. The dynamics of vapor compression cycle systems are dominated by the thermal behavior of the heat exchangers that have the slowest dynamics. Often these heat exchangers are modeled by partial differential equations (PDEs). Subsequent to developing the appropriate PDEs, the overall system is discretized into finite elements and a computationally intensive numerical calculation is performed to obtain accurate steady-state behavior [17]. The PDE approach to modeling a multiphase fluid heat exchanger is ill suited for most of the controller design tools currently available, particularly computer-aided control system design tools for multipleinput, multiple-output (MIMO) controller design. Therefore, it is beneficial to transform the PDE into a low-order ordinary differential equation (ODE) representation as a simplification and approximation for control-oriented modeling. This approach requires several assumptions about the fluid flow in the heat exchangers to simplify the coupled, nonlinear, PDEs given by the conservation of mass, momentum, and energy. These assumptions are as follows: ● Assumption 1: The heat exchanger is a long, thin, horizontal tube. ● Assumption 2: The refrigerant flowing through the heat exchanger tube can be modeled as a one-dimensional fluid flow. ● Assumption 3: Axial conduction of the refrigerant is negligible. ● Assumption 4: Refrigerant pressure along the entire heat exchanger tube can be assumed to be uniform. Assumption 4 indicates that pressure drop along the heat exchanger tube due to momentum change in refrigerant and viscous friction is negligible; therefore, the conservation of momentum equations are not needed. The PDEs that govern fluid flow can be found in most fluid mechanics textbooks [18]. By applying these assumptions, it is possible to simplify these equations to one-dimensional PDEs. A detailed explanation of these steps can be found in [19]. The resulting equations for fluid flowing through the heat exchanger tube are as follows: Conservation of mass:

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Gas Cooler

Compressor

Internal Heat Exchanger

Expansion Valve Evaporator (Receiver)

Figure 1. A diagram of the transcritical vapor compression cycle. The major physical components are shown. The dynamic responses of the compressor and expansion device are much faster than those for the heat exchangers. ˙ ∂(ρ A) ∂(m) + = 0, ∂t ∂z

(1)

Conservation of refrigerant energy: ˙ ∂(ρ Ah − AP) ∂(mh) + = pi αi (Tw − Tr ), ∂t ∂z

(2)

Conservation of tube wall energy: (C p ρ A)w

∂(Tw ) = pi αi (Tr − Tw ) + po αo (Ta − Tw ), (3) ∂t

where ρ = density of refrigerant, P = pressure of refrigerant, h = enthalpy of refrigerant, pi = inner perimeter (interior surface area per unit length), po = outer perimeter (exterior surface area per unit length), Tr = temperature of refrigerant, Tw = tube wall temperature, Ta = ambient air temperature, αi = heat transfer coefficient between tube wall and internal fluid, αo = heat transfer coefficient between tube wall and external fluid, A = cross-sectional area of the inside ˙ = refrigerant mass flow, and C p ρ A w = thermal of tube, m capacitance of tube wall per unit length. The governing PDEs are used to derive lumped parameter ODEs to model the dynamics of heat exchangers. The heat exchanger, which may be an evaporator, gas cooler, or internal heat exchanger, is divided into sections with a moving boundary between fluid phases. Furthermore, (l)–(3) are integrated along the length of each tube section. Figure 2 shows a typical condition for the evaporator with the fluid entering as two phase and exiting as a superheated vapor. The evaporator is modeled with two regions: a two-phase region and a superheated region. The boundary between these regions is a moving interface, which is difficult to measure physically.


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oriented model is a lower order one that still provides physical insight. In [19], a singular perturbation analysis showed that the dynamics of the energy stored in the refrigerant can be neglected while maintaining a majority of the dynamical information. Singular perturbations were used rather than balanced truncation so as to retain physics-based system knowledge in terms of known states such as temperature and enthalpy. References [19] and [21] detail a numerical comparison based on singular values and eigenvalues of the original and reduced-order models of the system. After an extensive iterative modeling, simulation, and validation process, it was determined that the dynamic system order was essentially equal to the number of individual heat exchanger wall sections shown in Figure 2 because these dominated the dynamics of energy storage and release. Figure 5 shows a comparison of the 11th-order nonlinear model with reduced order models and the actual data. As can be seen in Figure 5, the reduced-order models do a good job of reproducing the system’s dynamics. The maximum absolute value of the time-domain model error in Figure 5 is less than 10%, well within acceptable tolerances for feedback control design. It should be noted that other input–output model pairs were not as accurate as Figure 5 [19] but all were within 20% error when examining the maximum absolute value of the timeQuality = 1 domain model error for many different PRBS Twall,2(t ) signal amplitudes.

Details on the development of models for the evaporator and other components in the system can be found in [19]–[21]. This procedure is similar to the work of [22] that examined the dynamics of subcritical cycles. The resulting system models consist of 11 ODEs and several calibrated algebraic relationships. These include nonlinear dynamic models as well as linearized versions. The validity of the modeling approach on this system can be seen in Figure 3. The data for Figure 3 were obtained from a prototype automotive transcritical air-conditioning system [23] operating at highway conditions. This test system is part of an NSFsponsored Air Conditioning and Refrigeration Center at UIUC; a full description of the experimental system can be found in [24]. As shown in Figure 3, the model is accurate in predicting the dynamics of the system given the correct parameters. Although the 11th-order model does a good job of describing the system dynamics, a time-domain system ID test reveals a significant amount of over-modeling in the process. Figure 4 shows the results of a pseudorandom binary sequence of pulse inputs given to all inputs of the system along with a fifth-order input–output model derived with system identification tools. The inputs varied are the expansion valve opening, compressor speed, and both heat exchanger fans. It is clear that the most control-

Twall,1(t ) •

minhin Qualityin > 0

P(t )

•

mouthout(t )

Discussions on Lesson 1

In achieving these models, several assumptions and compromises were made. A primary focus was to retain the physics-based repreL2(t ) L1(t ) sentation rather than a numerical balanced LTotal truncation approach to the reduced models. In Figure 2. An evaporator with a two-phase flow at the entrance and supergeneral, the choice of assumptions rests with heated vapor at the exit. A conceptual moving boundary separates the two the user and the particular process. This fluid phases and dictates the heat transfer characteristics along the tube. choice involves a careful and conscious tradeoff between accuracy and simplicity. A low-order model may 3,460 have little use for a thermal systems 3,440 design engineer, but it may be appro3,420 priate for a control engineer. Howev3,400 er, referring to the begining of this 3,380 article and [11], 80% accuracy is 3,360 often sufficient in the model with Data Nonlinear Model feedback compensating for the 20% 3,340 Linear Model uncertainty. Therefore, a tradeoff to 3,320 0 50 100 150 200 250 300 350 400 forego some accuracy can be advanTime tageous for control. It is important to note that this Figure 3. Evaporator pressure for step changes in compressor speed. This close fit control-oriented modeling example between the experimental and simulation data demonstrates good model accurawould be practically impossible to cy. A linearized model provides results similar to the more detailed nonlinear achieve without the experimental model for this operating condition. Single Phase

Pressure [kPa]

Two Phase

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Superheat

5

Measured Output Fifth-Order Fit: 79.63%

0 −5 0

200

400

600

800

1,000

1,200

1,400

Evaporator Pressure

500 0

Gas Cooler Pressure

−500 0 200

Evap Exit Air Temp

200

400

600

1,000

800

1,200

1,400 Measured Output Fifth-Order Fit: 76.87%

0 −200 0 2

200

400

600

1,000

800

1,200


0 −2 0 1

GC Exit Air Temp

Measured Output Fifth-Order Fit: 87.3%

200

400

600

1,000

800

1,200


0 −1 0

200

400

600

1,000

800

1,200

1,400

Figure 4. System identification results for a pseudorandom binary input sequence. All system inputs (compressor, valve, and two fan speeds) are varied to get a good MIMO identification. Model matching indicates that a fifth-order overall system dynamic model is accurate.

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3,460 3,440 Pressure [kPa]

apparatus [23], [24] that was continually used to verify the model validity. A detailed physics-based knowledge of the process under study will help in making the simplicity/ accuracy tradeoff, and understanding the algebraic or dynamic relationships of the system is best done in conjunction with experimental data. Assumptions can be made and tested in simulation and then compared with experimental data to determine whether they are valid or not. Detailed nonlinear maps relating various fluid properties, valve characteristics, and compressor efficiencies can be found and refined best with an experiment to assist the

3,420 3,400 3,380 Data Nonlinear Model 11th-Order Linear Model 5th-Order Linear Model

3,360 3,340 3,320

0

50

100

150

200 Time

250

300

350

400

Figure 5. Evaporator pressure for step changes in compressor speed. The fifthorder linear model is capable of capturing the major system dynamics well, thereby justifying a reduced-order modeling approach.


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process. Although it is relatively easy to determine control-oriented models for simple mechanical linkage systems driven by dc motors (e.g., robots), we contend that experimentation is a necessary condition for successful control-oriented model development in complex systems such as a vapor compression cycle.

Importance of a Well-Defined Control Objective The experimental system studied here is a half-car active suspension as shown in Figure 6 [25]. This experiment falls under the category of a control technology experiment because the appropriate hardware has been chosen by industrial practice. Despite initially failing from an economic justification viewpoint, which is another valuable lesson learned, active suspensions provide an excellent case study in the appropriate choice of control objective. Although there has been a wealth of information published

Figure 6. The UC Berkeley half-car active suspension. A fullscale testbed involving actual vehicle components is used for investigating controllable suspensions.

ms

zs ks

u

mu

bt

zu

kt

zr

Figure 7. A quarter-car active suspension schematic. A 2DOF linear system model assumes damping in the tire and an ideal actuator between the wheel and car body.

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on the control of quarter-car, half-car, and full-car suspensions, few of these works have incorporated the dynamics of the actuator in their analysis. As a result, most of these published works have been relegated to simulation studies, with a few notable exceptions [26]–[29]. In addition, all of the experimental exceptions were limited in their closed-loop bandwidth. As it turns out, the actuation is a crucial component in the overall systems analysis. Due to power requirements, coupled with packaging constraints, electrohydraulics are the primary practical choice for a fully active system. However, the dynamics of electrohydraulic systems pose a challenging problem for standard formulations of active suspensions. Most standard quarter-car formulations of the active suspension problem consider a system such as that shown in Figure 7, where the system input is assumed to be a force. This regulation problem is often posed in an optimal control framework such as a linear quadratic regulator [30]. The states are the suspension deflection (z s − zu ), tire deflection (zu − xr ), as well as the sprung and unsprung mass velocities. A review of previous optimal control strategies applied to the active isolation problem can be found in [31]. The resultant control input is a desired force applied to the system, and this force is a function of the system states. This control law usually involves an inner force control loop applied directly to the actuator, possibly with some force measurement to act as a feedback as described in [27]. The problem is that electrohydraulic actuators are fundamentally limited in their ability to track forces of any reasonable bandwidth. As explained in [32], typical electrohydraulic systems are limited in their ability to do force control when interacting with an environment possessing dynamics. For a typical electrohydraulic system, the poles of the plant with which it is interacting will manifest themselves as the zeros of the open-loop force transfer function. If these zeros are lightly damped, the achievable bandwidth of any controller will be limited. Given the lightly damped modes for a typical quarter-car system of Figure 7, the force loop zeros will usually be lightly damped. Therefore, the use of this actuation to generate a controllable force for active suspension isolation can be said to be an ill-posed problem because the loop that any controller will be trying to close will have zeros near the jω axis that fundamentally limit the performance of any feedback algorithm. This difficulty is evidenced by the dearth of literature on successful experimental active suspensions using electrohydraulic systems. Additionally, nearly all results to date [26]–[28], including production prototypes [29], are limited to relatively low-frequency road disturbance rejection because of the bandwidth limitation. The primary problem with many of the earlier active suspension investigations was that the control objective was not properly defined. If the control objective is formu-


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••

zu

Gref (s ) = −

s(

+ 2ζωns +

(zs−zu)output

(zs−zu)reference

s + 2ζωn s2

)

ω2n

Gplant(s)

Gcontrol(s)

+ −

Inner Feedback Loop

Figure 8. A block diagram schematic of problem reformulation. The emulation problem has been recast as a reference tracking problem. The prefilter specifies the closed-loop suspension characteristics. lated in terms that are favorable to the actuator dynamics, then the problem becomes much easier. As electrohydraulic systems controlled with directional valves can be approximated as an integrator from input to position output [33], a natural formulation would involve one of velocity or of position tracking. The key to successful problem reformulation is the choice of an appropriate reference system that the system of Figure 7 should emulate. One effective reference system is the inertial damping approach, or skyhook damping, popularized by Karnopp in the 1970s [34]. Considering the system shown in Figure 7, if the controllable actuator were to emulate an inertial damper, the resultant system would appear as follows: The transfer function between the unsprung mass acceleration, and the suspension deflection is z s − zu s + 2ζ ωn ≡ G ref (s), =− 2 s(s + 2ζ ωn s + ωn2 ) z¨ u

(4)

√ where ωn = ms/ks and ζ = (b/2 msk). If the control system is able to make the actuator track this desired position, then the overall system will behave as if it were actually inertially damped. Figure 8 captures the essential nature of the reformulated control objective. What was often posed in the active suspension literature as a regulation problem is reformulated as a tracking problem in which an appropriate prefilter defines the actuator displacement reference signal for the control loop to track. The tracking problem is much easier to solve because the fundamental limitations associated with the force control have been eliminated. It should be carefully noted that the reformulation illustrated in Figure 8 is not specific to the linear inertial damper reference system of Figure 9. Any suitable reference model could be chosen as long as a well-posed relationship exists between unsprung mass acceleration and actuator displacement. Figure 10 shows the comparison of an ideal inertially damped system with an experimental system utilizing the reformulation shown in (4). The details of the experimental system, which is a quarter-car analog, are given in [35]. For the inner feedback loop of Figure 8, a model reference adaptive controller [36] was implemented. Figure 11 shows the normalized · 2 ratio of

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disturbance and sprung mass acceleration as a function of frequency. Clearly, the experimental performance is very close to the ideal. Figure 11 demonstrates excellent broadband vibration isolation across a relatively wide range of frequencies from 0 to 20 Hz. This meets necessary specifications to cover the 0–15 Hz frequency band that usually spans the two primary modes of the suspension system in Figure 7: approximately 1 Hz and 10 Hz. This is even more interesting when considering that few experimental active suspension strategies with electrohydraulic actuators have been able to perform at any level above approximately 5 Hz.

Discussions on Lesson 2 An ill-posed active suspension control problem with electrohydraulic actuators can be made well posed by reformulating it in a tracking framework. The experimental aspect to the control system goes beyond the idealized system of Figure 7 and helps understand the discrepancy between the theoretically achievable performance and that obtained in practice. Without the experimental aspects, it would have been difficult to develop an explanation regarding lightly damped performance-limiting zeros and then reformulate the problem to eliminate them. ln essence, the inclusion of experimental efforts into the research allowed us to develop and refine a well-defined control objective.

b ms

zs ks

zu

mu bt

kt

zr

Figure 9. An inertially damped quarter car. This passive system is emulated by the closed-loop active suspension system. Proper emulation makes the vehicle body feel as if it were damped with respect to a fixed reference.


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Acceleration RMS Ratio

advanced multivariable control of offhighway power trains. Details on the power-train system and the coordination objective can be found in [37]–[39] along with MIMO H2 and H∞ control designs. This system could be classified as a control validation experiment for the reason that the EVPS was intentionally designed as an abstraction of an earthmoving vehicle’s power train. Several physical aspects were left out in the actual design. One example is the ac motor that is controlled to emulate either a spark ignition or compression ignition engine in real time [38], [39]. Other examples are the load subsystems. In lieu Figure 10. An electrohydraulic powertrain schematic with a load emulator. The of running an actual vehicle in an offhydraulic motor/pump combination emulates a physical load coupled to the power train. The electric motor is controlled to behave as an internal combustion highway environment, controllable engine, and the variable displacement pump acts as a variable transmission. loads emulated by using a pressure relief valve were imparted to the hydraulic motors representing the steering, drive, and working implement systems of the Appropriate Actuation and vehicle. The working implements on a wheel loader examSensing Authority This section focuses on actuation of a load emulator for ple would be lift and tilt functions. Cost concerns were an earthmoving vehicle power-train testbed. An earthmov- the major factor in the design choice for loading. More ing vehicle power train is a MIMO fluid power system, in details on the loading system can be found in [40], but which the power distribution needs to be coordinated. key ones will be given here. We consider only one of the three load loops for the With multiple loads competing for the limited total flow and available power in a hydraulic transmission, current purpose of exposition. It can be schematically represented machines rely on human coordination for different tasks. as shown in Figure 10. The ac motor, which is controlled to Active power-train control can be one means to achieve behave as an internal combustion engine, drives a variable better performance and less dependence on humans in displacement pump that sends high-pressure fluid through the loop. As part of the Caterpillar Electromechanical Sys- a controllable valve orifice. The flow valve shown in Figure tems Laboratory, an earthmoving vehicle power-train sim- 10 can modulate the flow to a hydraulic motor. The motor ulator (EVPS) was constructed at UIUC as a testbed for is mechanically coupled with a gear pump, cycling hydraulic fluid in a load loop hydraulically independent of the main loop. The load-loop pressure acts as a motor resistance and needs to be controlled by a pressure relief 1 Ideal Ratio valve, termed the load valve, to emulate different load 0.8 Experimental Ratio dynamics. The choice of a hydraulic loading system, rather 0.6 than an electrical one, was largely due to cost. Eddy-current dynos or ac motors necessary to generate load 0.4 torques and absorb 25 kW on each load node are much 0.2 more expensive than the simple gear pump/valve combination. However, less expensive components come with per0 0 5 10 15 20 formance tradeoffs that should be evaluated up front in Disturbance Frequency [Hz] the experiment design process. The load valve actuator used to generate the hydraulic Figure 11. Transmissibility ratios. The closed-loop emularesistance is a two-stage pressure relief valve whose chartion strategy allows the experimental system to behave like acteristics are relatively insensitive to flow change. The an ideal inertial damper over a broad range of frequencies. goal in this case is to emulate an actual load as if the The high-frequency deviation from ideal performance is due to actuator bandwidth limits. hydraulic gear motor were part of a real earthmoving vehi-

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cle. As the load valve can only absorb power, however, it depends on the output speed of the hydraulic gear motor to generate a desired load. This objective has been cast [41], [42] into a unique control framework that we term resistive control. The performance of the controller is solely dependent on its ability to act as a resistive load, with the assumption being that the engine supplies enough power such that part of it can be dissipated. A generalization of the resistive framework is shown in Figure 12. We define the driving subsystem as the source and the driven subsystem as the load. Figure 12 shows a source system G S under an actual load G L . Within the context of this discussion, the source system represents the wheel motors and the load system includes the environment (rolling resistance, road grade plus vehicle weight, and bearing friction) that provides resistance to the motion of the wheel motors for this earthmover example. The driving signal yd (wheel motor speed) and the loading signal yr (load torque on wheel) define the boundary between the source and the load. dS is an exogenous signal into the source, such as the net flow coming from the main pump into the motor. The goal of the controller is to emulate the actual load G L by shaping the dynamics from dS to yS so the source (i.e., wheel) thinks it is attached to an actual load (i.e., environment). For a controller to manipulate the closedloop dynamics from dS to yS , it can use dS as a feedforward signal and yS as a feedback signal, assuming the controller has access to both measurements. The following discusses the controller taking only yS to generate the loading signal, creating a one degree-of-freedom (DOF) feedback design. A two-DOF design using both dS to yS with both feedback and feedforward elements can also be considered as in [41]. If possible, it is more desirable to use only the driving signal yd as shown in Figure 13, where the components are specific to this earthmoving vehicle example. In this case, the interconnection of the source and the emulator represents that of the source and an actual load. G a represents the actuator dynamics. The main idea is to emulate the load, that is, to let K ∗ G a = G L , so that the closed-loop dynamics will match the reference system. When this approach works, it is the most straightforward way to emulate a load and is convenient for emulating complex loads, nonlinear loads, or loads with time-varying parameters because the controller preserves the original representation of the actual load. As detailed in [41], it is necessary for the actuator to be significantly faster than the desired closed-loop bandwidth of the emulator. Figure 14 shows time- and frequencydomain results for both a direct and a resistive control approach for load emulation. In contrast to a resistive approach, the direct controller assumes that a torque signal is the controller output, and this torque can be directly

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applied to the system regardless of the reference input. Direct control would occur if the pump and valve of the loading loop in Figure 10 were replaced by a direct drive ac motor with associated power electronics and cooling circuits. Both direct and resistive controllers assume the same controllable actuation bandwidth, and both utilize an H∞ control design procedure [41]. As seen in the frequency responses for direct control in Figure 14, where an arbitrary input can be applied independently of the external reference, the closed-loop system performance rolls off as the actuator loses bandwidth. However, for the resistive control approach associated with the load emulation, the closed-loop response approaches the open-loop plant response at high frequencies. This indicates that the closed-loop system becomes independent of controller design and actuator bandwidth at high frequencies. This phenomenon can also be seen in the time-domain responses of the system. For direct control, the closed-loop system response can be made to track the reference model speed throughout a step change in driving torque. For the resistive control, the speed briefly tracks the open-loop model at the start of the step change in input torque and then settles down to track the reference model. The initial response of the resistive controller indicates that the system behaves

Figure 12. The concept of source and load in resistive control. The load affects the power source that, in turn, affects the generation of the load. In a resistive case, the load can affect only the rate of energy dissipation and cannot independently add energy. A detailed explanation can be found in [41].

Figure 13. One-DOF feedback load emulation. Direct measurement of the source output is used by the controller to set the prescribed load on the source system. The controller contains a model of the load dynamics in addition to other information.


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like the open-loop dynamics for high-frequency signals. The eventual convergence to the reference model indicates the resistive approach tracks well for lower frequencies. The unique phenomena illustrating the convergence of the closed-loop behavior to the open-

EVPS experimental load controller. A specific control problem for an experimental apparatus thus resulted in a basic and fundamental insight into a unique aspect of how actuation bandwidth limitations affect closed-loop system dynamics. Typically, one assumes that actuator rolloff implies a rolloff in closedloop behavior. In this case, however, actuator rolloff implies convergence to open-loop plant behavior. By understanding the underlying phenomenon, it is possible to appropriately specify the valve bandwidth necessary to achieve a desired emulation task. In doing so, the costs associated with the load aspect of this control validation experiment could be reduced by an order of magnitude over competing technologies while still meeting the system performance specifications based on the maximum load dynamics bandwidth.

Control technology experiments occur when a physical system, designed and built a priori, needs to be controlled and is presented to the researcher warts and all. loop dynamics in Figure 14 is specific to the class of resistive systems such as the EVPS load emulator in Figure 10, many types of engine test dynamometers [44], and other hardware-in-the-loop testbeds [45].

Discussion on Lesson 3 A control engineer should thoroughly understand how the choice of actuation and sensing methods affects the closed-loop performance. In the case of resistive systems, an understanding of how the indirect nature of the actuation affects the high-frequency system behavior would not have been discovered had it not been for the

Serendipitous Discovery and New Paths

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Magnitude

Speed Output [rad/s]

Speed Output [rad/s]

Magnitude

This section studies on-road automotive vehicles. Unlike other investigations into vehicle control that take a control technology experiment viewpoint [46]–[49], we will detail a control validation experiment viewpoint. For the studies given in [46]–[49], full-sized vehicles were instrumented and interfaced. This approach involves a large capital H∞ Design of Direct Control H∞ Design of Reseistive Control investment in the vehicles, control Open Loop GS electronics, and testing facilities. ↓ Open Loop GS ↓ 100 100 Additionally, safety concerns need to be addressed in the full-sized vehicle ↑ Closed Loop GSK Reference Model GSL→ experiments. To circumvent the cost Reference Model GSL→ 10−2 10−2 and inherent danger in testing aggressive vehicle controllers using Closed Loop GSK → full-sized vehicles, a scaled vehicle Actuator Model Ga → Actuator Model Ga → 10−4 10−4 testbed was developed as an evalua10−3 100 103 10−3 100 103 tion tool to bridge the gap between Frequency [rad/s] Frequency [rad/s] simulation studies and full-sized 3 3 hardware. This testbed is known as the Illinois roadway simulator (IRS), Open Loop GS↑ Open Loop GS ↑ shown in Figure 15. Previous scaled 2 2 vehicle experiments, such as [50], Reference Model GSL Reference Model GSL had mainly involved moving the (Dashed) ↓ (Dashed) ↓ 1 1 vehicles along a fixed surface, which naturally incurs a host of interfacing (Solid) (Solid) ← Closed Loop GSK ← Closed Loop GSK and sensing issues. The IRS moves 0 0 0 1 2 3 4 0 1 2 3 4 the road while the vehicle is held Time [s] Time [s] fixed with respect to inertial space, which simplifies interfacing and Figure 14. Performance comparison of direct control and resistive control. At implementation. The design inspirahigh frequencies, the dynamics of the resistive controlled system converge to opention for the IRS came from comparaloop plant dynamics. For direct control, the closed-loop dynamics roll off with the tive locomotion studies done by actuation bandwidth at high frequencies. IEEE Control Systems Magazine

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biologists on different types of animals [51]–[53]. Further details on the IRS design and construction can be found in [54] and [55]. The IRS is a control validation experiment because careful design eliminated many of the details associated with an actual vehicle. There is no internal combustion engine, and accurate inertial sensing is readily available. Additionally, the dynamics of the vehicle are confined to a plane because the pitch, roll, and heave modes have been designed out. Similar to motivations stated earlier, the goal of the IRS was to provide a physical system to demonstrate particular control algorithms. The system was designed to make it easier to illustrate these concepts. There are several vehicle control concepts that have been developed with the IRS. These include a four-wheelsteer (4WS) or differential wheel torque approach we term driver-assisted control (DAC) [55]. The goal is to utilize the rear-wheel steering or torque differential between left and right wheels to give the driver a specified response from the steering input to vehicle yaw rate. As such, the vehicle can attain any handling characteristic within physical constraints such as tire-road friction. Subsequent to developing controllers and implementing them on the IRS, it was imperative to justify that the results obtained were valid on a full-sized vehicle. To justify the validity of the controller design and the corresponding results, it was necessary to put the model and controller structure into a framework that is independent of physical size. This led to the use of dimensionless analysis [56]. The ability to compare scale experiments to full-sized vehicle experiments relies primarily on the concept of dynamic similitude. The Buckingham Pi theorem [57] provides a useful tool to study dynamical systems in a dimensionless framework. It states that the solution to a differential equation can be made invariant with respect to the dimension space spanned by the parameters in the differential equation. This property is exploited through nondimensionalizing the differential equation by grouping the parameters into (n − m)-independent dimensionless parameters called pi-groups, where n is the number of parameters and m is the dimension of the unit space occupied by the parameters. For two systems modeled by the same differential equations, the systems are dynamically similar if the pi-groups associated with the differential equation are numerically the same for both systems. Application of the Buckingham Pi theorem to the classic bicycle model vehicle dynamics yields the pi groupings 1 =

a , L

b , L C αr L , 4 = mU 2 2 =

C αf L , mU 2 Iz 5 = 2 , m

Figure 15. The IRS system: a mechatronic analogy to the wind tunnel for scaled aircraft. The vehicle remains fixed with respect to inertial space while the roadway moves.

3 =

(5)

where m = vehicle mass, Iz = vehicle moment of inertia,

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V = vehicle longitudinal velocity, a = distance from the center of gravity (C.G.) to front axle, b = distance from C.G. to rear axle, L = vehicle length, a + b, C αf = cornering stiffness of front two tires, and C αr = cornering stiffness of rear two tires. Derivation and explanations of the vehicle pi parameters are given in more detail in [55] along with the dynamic similitude analysis proving similarity between IRS and fullsized vehicles. There are key advantages to examining the vehicle dynamics in the pi-space versus a dimensional state or parameter space. It is well known that the vehicle dynamics change significantly with both velocity and cornering stiffness. In [58], it is shown that for a given vehicle, 4 is

Figure 16. Parameter root loci. A nondimensional dimensional framework illustrates a duality between two different parameter variations. Cornering stiffness and longitudinal speed have the same effect on vehicle dynamics.


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usually proportional to 3 . Therefore, both velocity and cornering stiffness variations correspond approximately to variations in a single parameter, namely, 3 . This is shown in Figure 16 where the open-loop system eigenvalues are shown with respect to the two different parameters: velocity (represented by “o”) and cornering stiffness (represented by “x”). For the variation of the system roots with respect to cornering stiffness, the ratio of front to rear cornering stiffness was assumed to remain the same.

dynamics and control framework. This serendipitous discovery process arises from experiments and theory working together. The idea of a dimensionless framework has led to other avenues of research, including the apparent duality between cornering stiffness and vehicle velocity, a phenomenon that was qualitatively understood but not previously formalized. This discovery process is one of the key benefits to working with experiments.

Conclusions This article is meant to illustrate some of the key lessons to be learned by combining experiments with control theory and analysis. To demonstrate the breadth of application, different types of control experiments were used to illustrate each lesson. Neither the lessons nor their exposition were exhaustive, but they did represent a core subset of ideas that were identified by over 70 representatives from industry and academia during an intensive two-day workshop focused on automotive systems. It is likely that these core ideas could be applied in a variety of other fields. It is hoped that this exposition provides the reader with concrete examples they can apply to their own research investigations. Moreover, we hope this will encourage others to generate their own list of best practices or lessons learned that they can share with the research community. There are only positive aspects to the combination of experiments with control theory. Certainly experiments can be expensive to create and maintain; this includes both capital investment as well as personnel. It may be tempting to insist that cheaper simulations can do the job of demonstrating controller design concepts. However, that would be missing a large part of the picture. One only gets out of a simulation what one puts in. By involving actual hardware, the researcher is engaged in a continuous cycle of discovery.

We present four basic lessons learned from combining control theory and experimental implementation. These plots, similar to a root locus for controller design, demonstrate that the effect of road-friction variations on the underlying vehicle dynamics is dual to the effect of velocity variations. Therefore, these system parameter variations can be combined. By examining the system in a dimensionless framework, we were able to see the duality between velocity and road-friction. The use of the IRS as a control validation experiment was vital in directing us to consider these scaling issues that then led to a new discovery associated with the basic dynamics of vehicles. The experimental system also provided a motivation for new research areas in the control of systems that vary over length scales. Motivated by the IRS results, previous work by [59] examined the concept of robust control within a dimensionless framework. The goal was to provide a robust control algorithm that was independent of the length scale of the plant. Therefore, if the plant changed in dimension, the same control law could be used as long as it was appropriately scaled. Reference [58] details a linear matrix inequality-based robust control approach where ranges are placed on the pi parameters based on a distribution of vehicles examined from the literature. In [59], a frequency domain H∞ controller is designed for vehicle lane tracking in a dimensionless framework. The multiplicative uncertainty bound for a data set of over 50 representative vehicles is found to be much tighter, and hence less conservative, with the system description given in the dimensionless framework. This observation leads to less conservative robust controllers that are able to stabilize any vehicle in the data set while achieving the prescribed performance specifications.

Discussion on Lesson 4 The justification of what was originally a vehicle control testbed led to the investigation of a dimensionless vehicle

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Acknowledgments The support of Caterpillar Inc., Ford Motor Company, NSF, ONR, the University of Illinois, and all the ACRC member companies was instrumental in developing the results presented here. This is greatly appreciated.

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Coordinated Science Laboratory, University of Illinois, Urbana-Champaign, 140 MEB, MC-244, 1206 West Green Street, Urbana, IL 61801, U.S.A., [email protected].

Sean Brennan received B.S. degrees in mechanical engineering and physics from New Mexico State University, Las Cruces, in 1997. As a National Science Foundation Graduate Fellow, he received the M.S. degree in 1999 and the Ph.D. degree in 2002 in mechanical engineering from University of Illinois, Urbana-Champaign. He is currently an assistant professor at Pennsylvania State University in the Mechanical and Nuclear Engineering Department. His theoretical interests include unifying the concepts of scaling and control theory in the areas of robust and adaptive control, with application interests ranging from vehicle chassis control, to mechatronics, to MEMS.

Bryan Rasmussen received the B.S. degree in mechanical engineering magna cum laude from Utah State University, Logan, in 2000 and the M.S. degree in mechanical engineering from the University of Illinois, Urbana-Champaign, in 2002. He is currently pursuing the Ph.D. degree from the University of Illinois. His current research focus is on dynamic modeling and control of thermo-fluid systems.

[57] E. Buckingham, “On physically similar systems; illustrations of the use of dimensional equations,” Phys. Rev., vol. 4, pp. 345–376, 1914. [58] S. Brennan, and A. Alleyne, “Robust scalable vehicle control via non-dimensional vehicle dynamics,” Vehicle Syst. Dyn., vol. 36, no. 4–5, pp. 255–277, 2001. [59] S. Brennan and A. Alleyne, “H-infinity vehicle control using non dimensional perturbation measures,” in Proc. 2002 American Control Conf., Anchorage, AK, pp. 2534–2539.

Andrew Alleyne received the B.S. degree in mechanical and aerospace engineering from Princeton University in 1989 and the M.S. and Ph.D. degrees in 1992 and 1994, respectively, from The University of California at Berkeley. He joined the Department of Mechanical and Industrial Engineering at the University of Illinois, Urbana-Champaign, in 1994 and was appointed to the Coordinated Science Laboratory. He currently holds the Ralph M. and Catherine V. Fisher Professorship in the College of Engineering. His research focuses on modeling and control of nonlinear mechanical systems and is a mixture of theory and implementation. He is currently an associate editor of the ASME Journal of Dynamic Systems, Measurement and Control, and IEEE Control Systems Magazine, as well as a co-editor for Vehicle System Dynamics. He can be contacted at M&IE Department and

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Rong Zhang received the B.S. and M.S. degrees in 1996 and 1998, respectively, from the Department of Automotive Engineering at Tsinghua University, Beijing, China. He received the Ph.D. degree in 2002 in mechanical engineering from the University of Illinois, Urbana-Champaign. His doctoral research was on multivariable robust control of nonlinear systems with application to hydraulic power trains. He joined General Motors R&D and Planning in 2002 as a senior research engineer. He is currently working on control theory technology for the next generation vehicle power trains.

Yisheng Zhang received the B.S. degree from Beijing Institute of Technology, Beijing, China, in 1996 and the M.S. degree from Tsinghua University in 1999, both in vehicle engineering. He received the Ph.D. degree in mechanical engineering from the University of Illinois, Urbana-Champaign, in 2003. He joined Eaton Corp. in 2003 and is currently working on hybrid electric and other advanced vehicle powertrains.


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