Resumen de Robust control strategies based on fractional calculus for robotic platforms
Control engineering has dealt with uncertainty since its inception. In fact, something so fundamental such as the concept of negative feedback, emerged as a solution to an uncertainty problem.
During the development of the first telephone lines, electronic amplification devices had a strong non-linear behavior. That was not a problem for the telegraph lines, that use constant signals, because it can be solved with a simple gain adjustment. But voice signals traveling through the newly developed phone lines, were fluctuating and uncertain, resulting in wave distortion due to non-linear amplification. Harold Stephen Black then proposed applying negative feedback to electronic amplifiers to improve the tracking of the input reference. As a result, the amplifier output is linear to the input, and therefore it is possible to boost the signal with much less distortion.
Previous works had already discussed the feedback topic, but Harold Black was the first to propose negative feedback as a control method, and therefore is considered a pioneer in the field. To some point, the foundations of control engineering were built by Black, and later by Harry Nyquist and Hendrik Wade Bode, during those years in the Bell Telephone Laboratories.
In this case, the use of negative feedback made possible to address the problem of uncertainty in the input and disturbance signals. This method is still today the main solution to input uncertainty, but this is only one of the uncertainties existing in a control system. The other two uncertainty problems found in control engineering are modeling uncertainties and noise. The scope of this thesis falls into the second problem: model uncertainty.
Most control strategies are based on the analysis of a system model, but the concepts of uncertainty and modeling are closely linked, like those of measurement and error. Absolutely all models are subject to uncertainty, showing a different degree depending on the system, from small uncertainties, such as an almost ideal simple pendulum model, to large uncertainties, as is the case of weather evolution. This modeling uncertainty can also be divided into two other groups. Systems that have a known structure but unknown parameters are usually classified under parametric uncertainty, to differentiate from the systems with unmodeled dynamics, usually due to non-linear behaviors.
Problem severity due to uncertainty is also different depending on the system considered, for example, weather forecasts work with large uncertainties, but the consequences of a prediction error are slight. However, a small modeling uncertainty in one of the systems in a nuclear power plant can have catastrophic results. Most control systems are halfway between these ends, therefore, some uncertainties may be neglected depending on the application, although they are always problematic.
For example, common uncertainty problems found in robotics and automation range from accuracy errors to system instability. All of them are undesirable, but the second group is much more dangerous than the first. Although stability errors are not catastrophic, they can lead to a good deal of damage, to the system itself and to the environment, including people in the area.
Therefore, many control approaches evolved to cope with uncertainty. As an example, without the aim of providing a complete list, the following control strategies pursue the solution of these problems:
• Feedback: Input uncertainties and disturbances.
• Stochastic: Signal uncertainties or noise.
• Robust: Model uncertainties and non-linearity.
• Adaptive: Time varying and non-linear systems.
Robust and adaptive techniques address the uncertainty of system parameters from different perspectives. Robust control consists of making the system insensitive to parameter variations using a constant controller while adaptive control addresses the variability of system parameter by changing the controller parameters in order to achieve a uniform final performance.
This thesis contribution is based on these two strategies, robust and adaptive control, from a fractional calculus perspective, usually known as fractional order control.
While in integer calculus the operators are integral or derivative, using the exponent to specify how many times the operator is applied, in fractional calculus the same exponents and operators are used, but the exponent is allowed to have any real number value. Therefore, concepts such as second order derivative are allowed, but one and a half order derivative is also possible.
This exponent generalization do not have still a known definite physical meaning, such as, for instance, the integer derivative, but this is not a problem as long as the mathematics remains coherent. In fact, the integer derivative is also no more than a concept, commonly used for real world modeling because it is consistent with reality, but still a concept.
Although this idea may look strange at first, there are many operators that have been generalized in mathematics, producing valuable results. For example, consider the ubiquitous power function.
Multiplication leads to power function as the notation of a repeated product. Letting the exponent to be a fraction, we find the concept of root, where the repeated product of this fractional exponent leads to an integer exponent. Similarly, repeated use of a fractional derivative operator results in an integer derivative operator, meaning that using just one of them must fall somewhere between the function and its derivative.
This particular class of operators are useful in control engineering, both in the modeling of physical systems, improving the models based on ordinary differential equations (ODEs), and also in the design of robust controllers, improving their performance compared to their integer counterparts.
On the one hand, modeling using fractional differential equations (FDEs) has been successfully used in the mathematical description of many processes such as heat transmission, flow diffusion, abnormal relaxation, and other physical phenomena, including engineering systems modeling.
On the other hand, robust controller design using fractional operators is known in the literature as fractional order control. The advantage of fractional controllers over integer controllers lies in their superior versatility. Just as fractional modeling with FDEs can describe behaviors that are impossible to model using ODEs, fractional controllers are able to implement filters that can shape plant characteristics in a way that integers cannot. Note that a controller can be seen as a filter applied to the error signal.
Hendrick Bode was the first describing a fractional order calculus application in robust control. In his words, a fractional transfer function model, would be able to maintain overshoot characteristics despite changes in the system gain. This feature is known today as iso-damping.
The tightness of integer controllers, hampers that robustness specifications in many cases, while their fractional counterparts can meet any robustness specification as their frequency response is much more varied and flexible. Nevertheless, there are still a lot of issues that need to be solved. There are many tools available for integer controller tuning, but unfortunately they are not directly applicable to fractional order control.
The problem of fractional controller tuning has been addressed many times from the early works to the latest proposals. Due to the nonlinear nature of the equations to be solved, a common problem in these proposals is the high computational complexity and the lack of information on the tuning process.
This thesis addresses that problem by using a graphical solution for the nonlinear equations, but the graphs are designed to be meaningful about the controller parameters, providing a great understanding of the tuning process. This provides a quick and easy solution for the class of controllers defined with a fractional operator known as fractional proportional derivative (FPD) and fractional proportional integral (FPI), which can be used in the robust control of any given system.
The main idea behind the proposal is the slope cancellation between the system phase and the controller phase. This approach allows an easy way to solve the robustness constraint, resulting in a simple math and a straightforward solution. Nevertheless, the resulting equation can not be solved for the controller parameters, as some variables can not be isolated. Then, a graphical solution for that equation is proposed.
A valuable process insight can be obtained using certain variables in the plot, offering an additional advantage. For instance, the graphical solution results in a set of curves having the same controller slope (iso-m). These curves can then be used to solve for the exponent, but also, the graph shows the needed exponent variation in order to widen or reduce the phase margin, or to widen or reduce the slope, allowing the fine tuning of the controller using just the graph and very simple math.
Although some works use already a graphical solution to avoid the computational complexity, the proposed plots do not offer information about the process and the graph is tailored for the combination of a controller and a plant, therefore, a new plot is needed for different plants. Moreover, an observer must visually spot the curve intersection, which impairs the computerization of the algorithm.
Unlike these other tuning methods found in the literature, this approach has a graphical nature that allows the designer to visualize the effects of parameter variations in a very simple and intuitive way. This is a very important contribution. Easy and fast controller tuning is possible simply using graphs or tables, without the need to solve a numerical nonlinear equation and allowing the solution of the control problem in a very intuitive and straightforward way.
This is important in applications that require changes in controller parameters during operation to optimize response towards a given target. In these situations, the method offers a fast and reliable solution, as the graphs can be used to directly determine the values of the controller parameters that meet the new control requirements. The proposed method allows not only to intuitively define the controller parameters that satisfy the design constraints, but also to observe at a glance how the parameters modify the loop dynamics.
Also, the graph can be tabulated, meaning that the entire process can be easily computerized. This provides a direct and unique tuning solution using very low complexity software capable of running on low cost/power boards and embedded hardware systems. This is a very interesting feature, useful in areas like robotics and automation, automotive industry, aerospace and many others.
In fact, the application of the iso-m method in a programmable logic controller (PLC) could be easily addressed, resulting in a good opportunity for industrial widespread of robust controllers, allowing to improve the quality of the current hardware systems.
Another issue found in the current fractional tuning methods is that non-linear equation solvers can only be used in off-line control schemes, due to the high computational effort and the high dependence on initial conditions and local minima. That makes their use in adaptive control strategies impossible.
Thanks to its straightforward solution, the problem of initial conditions and local minima is totally avoided using the iso-m method, providing both good performance and a reduced computing effort. The computational complexity of a lookup table is similar to a memory access, and although exact results are not achieved, any accuracy can be obtained by interpolation or by increasing the granularity of the table. Therefore, the iso-m method can be applied in adaptive control strategies, which leads to the second contribution of this thesis, an indirect fractional order adaptive controller.
Many works address this issue through a direct approach, which avoids the controller tuning, generally with a fractional reference model, but an indirect approach is not available to date. Although both strategies show good results, there are important differences in terms of implementation and applicability.
While direct strategies can sometimes result in simple equations that reduce computational complexity, they are limited to systems that meet certain conditions, and often adapt to them. For example, given that one of the implementation conditions is stability of both the plant and its inverse (minimum phase systems), the range of controllable plants that use this strategy is reduced, especially for discrete time systems. Also, performance must be specified using a reference model, therefore, even if the model is feasible, the controller required to achieve these specifications may not be. Although different solutions have been proposed to overcome some of these limitations, direct adaptive control cannot always be used.
Unlike the direct approach, indirect (or explicit) adaptive methods use current plant parameter estimates to determine controller parameters using a tuning method. This allows for a wide variety of control laws and plant estimation combinations. However, a previous system identification is always required. The general idea is based on an adjustable plant model to compare predicted and actual output. That difference is used by a model adaptation algorithm to minimize this error, so that a correct description of the plant is achieved asymptotically for the current sequence of inputs and outputs.
Once the plant parameter estimation convergence is solved, which is indeed a critical issue for indirect adaptation schemes, literally any tuning method based on a plant model can be used.
Given the high computational requirements of currently existing tuning methods, all fractional adaptive strategies are direct, using fractional reference models, but keeping integer order in adaptation algorithms and controllers.
This thesis describes and demonstrates how an indirect adaptive strategy can also be used in a fractional order controller. The key is that all tuning operations can be carried out within a sampling period thanks to the efficiency of the new proposed method.
The proposed adaptive fractional control uses real time plant parameters obtained through the recursive least squares (RLS) identification algorithm combined with the iso-m fast controller tuning method to join adaptability and robustness in a single control scheme.
Thanks to the robustness of the system, performance will not change for the operating point, and thanks to the adaptive scheme, plant changes will update the controller tuning to obtain the same original operating point specifications.
Therefore, this thesis proposes two new approaches to tackle the problems caused by uncertainty.
The first proposal is based on the use of robust fractional controllers, capable of delivering a robust response despite system mass changes and non-linearities. The presented method is applicable to any system and controller. The method focuses on controllers with one fractional operator, but the same approach can be used in other classes of controllers, with more operators, or different equations.
The second proposal is an unprecedented indirect fractional adaptive scheme that allows robustness to be extended to a wider set of systems, including time varying and nonlinear systems.
The simulation results show that despite introducing disturbances in the plant gains, the performance of the proposed controller is not only maintained, but also outperforms in results to other equivalent controllers. The iso-damping potential has been therein confirmed to ensure robustness to gain changes in the controlled plant.
The experiments performed show how the resulting system is able to cope with variations in plant parameters while maintaining short and long term performance settings. The systems used for validation are:
1. The Honda Accord autonomous car available for researchers in the Lawrence Berkeley National Laboratory (US). In this case, the iso-m controller shows a robust performance, rejecting possible disturbances on the high level response due to different dynamics, powetrains or road slopes. A hierarchical ACC control structure was used, with the fractional order controller in charge of the gap distance regulation through a reference acceleration tracked by a lower control layer. The results outperformed an equivalent integer order PD controller used for comparison.
2. The elbow of the TEO humanoid robot developed by the Robotics Lab team of Carlos III University of Madrid. The results obtained are very competitive compared to other well-known fractional order tuning methods found in the literature. The system performs according to specifications and shows high robustness to mass changes at the tip (robot hand), showing similar performances compared to other tuning methods of much higher complexity.
3. A bio-inspired neck made of soft material. Despite the non-linear and time varying properties of the soft material, the response of the inclination feedback control was totally correct, in accordance with specifications and showing high robustness to mass changes. Even for very high payloads compared to the weight of the neck (895.5% heavier), the system robustness is granted through the proposed iso-m controller. This system was a control challenge, due to the described plant properties, highlighting the benefits of the proposed methods in nonlinear time varying applications compared to other control methods.
The excellent results obtained in the real platforms show how easily it could be applied to any type of dynamics, which is convenient to encourage the adoption of fractional robust controllers in the field of automation. The robustness obtained in the performance of the three test platforms considered, confirm the suitability of the proposed methods for the robust control of LTV and NLTI systems.
To conclude, the robust iso-m controller tuning method can be applied to any real system. The only information needed for its application is the plant's frequency response for the desired specification. This can be obtained from a plant model, from system identification, or even within real-time adaptive control schemes.
The requirements are very low, both from the computing point of view and the mathematical complexity, therefore its application can be easily deployed in many industrial, embedded and low cost environments. There is no limit for its use in any control scheme.
Both iso-m and adaptive iso-m methods have proven to be a working solution to the major problem of uncertainty in very different platforms.
