The increasing complexity of control algorithms is an inevitable result of the gradually increasing performance requirements of control systems. On the one hand, as the performance requirements of control systems continue to rise, the number of factors that can affect the performance of the control system needs to be considered during the design of control algorithms (when the performance requirements are low, the impact of these factors on the final performance can be ignored). This necessitates a reasonable arrangement of the structure of the control algorithm to accommodate and handle these factors. On the other hand, most of the factors affecting the improvement of control system performance come from the physical/hardware level of the controlled system (specific processes/operations, the working characteristics of actuators, etc.). If some non-ideal factors cannot be avoided at the physical/hardware level (such as the hysteresis characteristics of piezoelectric crystals and hysteresis-stretching materials in precision motion platforms, flexibility and friction torque in transmission systems, nonlinearity of pressure drop-flow rate at hydraulic valve ports, dead time in temperature control systems, etc.), then the requirement to improve the performance of the control system must be entirely achieved by the control algorithm.
In fact, the design and performance improvement of control systems should be conducted within the framework of the entire system or product design. Control algorithm engineers should effectively collaborate with mechanical engineers, electrical/instrumentation engineers, process engineers, and other personnel to first mitigate problems and non-ideal factors at the system level. If the main factors affecting control system performance can be avoided or the control system performance improved by improving structural design/process flow, the control algorithm will need to handle fewer factors. Therefore, even relatively simple control algorithms can achieve high-performance control indicators. Conversely, if the physical/hardware design is unreasonable, no matter how much effort is put into the control algorithm design, it may not be able to meet the performance requirements (the characteristics of the controlled system determine the upper limit of its performance).
Therefore, the trend of increasing complexity in control algorithms should be approached rationally: While ensuring the problem can be solved, there should be no deliberate pursuit of complex control algorithms (if a PID controller/improved design can achieve the performance target, there is no need to use other control algorithms); when sophisticated and complex control algorithms are required, the controlled system should be carefully analyzed, the dominant factors affecting the performance of the control system should be summarized, a suitable mathematical model should be established, an appropriate control algorithm structure should be selected, and non-ideal factors at various levels should be addressed in a targeted manner, so that the problem structure and the control algorithm structure are matched.
The increasing complexity of control algorithms is reflected in three aspects: structure, (stability) analysis and design, and debugging and maintenance. To match the structure of real-world problems, the control algorithm itself needs to be adjusted in terms of structure. The complexity of the control algorithm's structure not only increases the number of design parameters but also enriches/complexes the dynamic behavior of the entire control system containing the control algorithm, making the stability analysis of the control system and the design of the control algorithm more complex. The complexity of the control algorithm's structure and design greatly increases the application threshold of the control algorithm, and the system debugging time and maintenance costs also increase accordingly.
Structural complexity of control algorithms
As shown in Figure 1, to address the numerous non-ideal factors in the controlled system, the design of control algorithms can evolve from the simplest PID controller commonly used in industry to a more complex structure (feedforward + feedback + state observation/filtering + coefficient/parameter identification). In practical applications, the control algorithm does not necessarily need to include all these modules simultaneously; rather, these modules are selected or combined selectively based on the specific problem structure.
Figure 1. Structural complexity of the control algorithm
The selection of the control algorithm structure is explained below, taking into account the non-ideal factors shown in Figure 1:
• Observation/filter modules are generally used to handle non-ideal factors in sensing systems. They can also compensate for unmeasurable external disturbances (such as load torque, like the load sensing function of a torque-less sensor in a collaborative robot) or parameter changes by constructing suitable disturbance sensors. Depending on the form of the non-ideal factors, observation/filter modules can be implemented using different algorithms: for example, Kalman filters are commonly implemented in motion control systems to generate velocity/acceleration signals from position signals, achieving phase-delay-free filtering; Luenberger/Kalman observers are particularly suitable for situations in industry where some states are unmeasurable or where sensors cannot be installed (if the system mechanism is clear and a relatively accurate system model can be established, then the observer is equivalent to constructing a virtual sensor, achieving accurate estimation of the unmeasurable), making full-state feedback possible; meanwhile, other types of observers, such as sliding diaphragm observers/high-gain observers, can perform state observation/disturbance compensation for nonlinear dynamic systems.
Similar to the observation/filtering module is the parameter identification module. This module accepts input/output data from the controlled object and identifies system parameter changes/external disturbance parameters based on the mechanistic model. The identified information is then fed into the feedback controller, enabling the control input to adapt to parameter changes. Additionally, the information identified by the parameter identification module can also be fed into the feedforward controller to achieve ideal compensation for unknown disturbances (such as nonlinear friction forces).
• Trajectory planning combined with feedforward control typically enables proactive responses to known or identifiable external disturbances (including input commands), improving the system's dynamic tracking performance. For underactuated systems (i.e., the degrees of freedom of the control input are less than the degrees of freedom of the output) or critically damped dynamic systems, effective trajectory planning (or input shaping) can further suppress system oscillations and ensure system control performance.
• The core mechanism of automatic control lies in feedback, and feedback controllers can effectively handle most non-ideal factors. If the system mechanism is unclear/cannot be accurately modeled/the model is too complex (such as chemical process control/eddy current field control), and sufficient data can be collected, then data-driven control methods can be used; if the system can be modeled relatively accurately, but there are model uncertainties/parameter variations/unknown external disturbances, then robust control (infinite H/sliding plate control, etc.) can be used; other control algorithms, such as adaptive control, model predictive control, feedback linearization, and other algorithms, as well as different combinations thereof, are designed specifically for different non-ideal factors.
Increased complexity in stability analysis and control algorithm design of control systems
The core characteristic of automated control systems lies in their ability to operate machines/equipment without manual intervention. Relying on real-time signals acquired and corresponding control algorithms, they provide dynamic control commands to enable automatic operation and meet expected performance requirements. The prerequisite for reliable automatic machine operation is the stability of the controlled system (comprising the controlled object and the control algorithm) – the system must not malfunction. If the stability of the control system cannot be guaranteed, the machine/equipment will not function properly, potentially leading to damage, injury, or even death. The structural complexity of control algorithms enriches the dynamics of the entire control system (depending on the specific situation, the entire controlled system may transform from a linear system into a nonlinear time-varying system, exhibiting multiple equilibrium points, attraction regions, finite escape times, and other nonlinear dynamic behaviors, increasing the risk of system instability). Therefore, stability analysis and the design of control algorithms based on stability analysis become particularly important.
The stability issue of a controlled system stems from the fact that it is a dynamically changing system. All variables characterizing the system's dynamics are constantly changing, ultimately leading the system to its expected operating point/range. If the control input is inappropriate (inappropriate control algorithm design) combined with the dynamic characteristics of the controlled object, certain variables may change far beyond the expected/reasonable range, entering an irreversible process (the system loses stability and cannot return to equilibrium or the expected state). For example, regarding the spread of COVID-19, without timely testing and tracing (measurement feedback) and corresponding mandatory and treatment measures (control input), the number of infections would increase exponentially, the healthcare system would collapse, and the epidemic would spiral out of control (system instability). Conversely, with timely testing and tracing, coupled with appropriate mandatory measures, the spread of the epidemic can be contained within a relatively small area (system stability) and ultimately brought under control.
In fact, depending on the research object (linear/nonlinear) and the scenario (the system is subject to external disturbances/changes in initial state/parameter changes/uncertainties, etc.), the meaning of stability and the corresponding criteria for judgment are diverse. These definitions of stability describe whether, at what rate, and within what range the variables characterizing the system's operating state converge as time changes, under different initial states and moments, and within what interval. The various definitions of stability are not merely abstract concepts proposed for theoretical analysis; they are closely related to the actual control performance achievable by a system. If a controlled system can be guaranteed to be globally exponentially stable, then regardless of the initial state, the system control error (or other variables) can be reduced to zero in a relatively short time and remain at zero for a long period. If only locally asymptotically stable stability can be guaranteed, then within a narrow operating range, the system control error gradually decreases to zero over time. If exponential/asymptotic stability cannot be guaranteed, and only bounded input/bounded output can be guaranteed, then it can only be expected that the system control error will converge within a certain range. In this sense, analysis based on stability can clearly define the upper limit of the performance achievable by a control system, and on this basis, a controller can be rationally designed.
Therefore, textbooks on advanced control algorithms (such as nonlinear control, model-based adaptive control, sliding mode control, or some data-driven control methods) often begin with numerous chapters on stability concepts and tedious stability analysis, only moving on to the specific design of the control algorithm later. This illustrates, on the one hand, the increasing complexity of control algorithm structures, which significantly enhances the stability analysis and design of control systems; on the other hand, it becomes a major obstacle for students or engineers wanting to learn advanced control algorithms. These stability analyses overemphasize various mathematical concepts, lacking discussion on the connection between these concepts and the real physical world. Furthermore, beginners often transition from classical control theory (based on transfer functions, single-input, single-output linear time-invariant systems with only a single equilibrium point, and system stability generally directly corresponding to global asymptotic stability) to the study of advanced control algorithms, lacking an understanding of the fundamental role of stability analysis in the design of advanced control algorithms. In reality, even if the controlled object is a single-input, single-output linear time-invariant system, the use of adaptive and parameter identification processes causes the overall closed-loop system to still exhibit time-varying nonlinear characteristics, and the system's stability will change (even if the original control system can guarantee stability). Understanding these stability analysis and design methods is essential for the successful implementation of advanced control algorithms.
Summarize
The increasing complexity of control algorithms arose to address the trend of continuously improving performance indicators of control systems. This is to enable targeted handling (methods include identification, adaptation, elimination, compensation, and suppression of influences) of various non-ideal factors affecting the performance improvement of control systems.
The trend towards increasing complexity in control algorithms should be approached objectively and rationally. We should not deliberately pursue algorithmic complexity (simpleness is better, provided the control system performance is guaranteed), nor should we shy away from the demands of real-world problems for more complex control algorithms. We must carefully analyze the controlled system, summarize the dominant factors affecting control system performance, establish a suitable mathematical model, select an appropriate control algorithm structure, and address non-ideal factors at various levels in a targeted manner, ensuring the problem structure matches the control algorithm structure. The design of the control system and control algorithm should be approached from the perspective of the entire system/product, emphasizing collaboration among engineers of various subsystems. Problems solvable at the system/design level should be prioritized for system-level solutions. This also requires supervisors/system engineers to possess basic concepts and learning and communication skills in their respective professional fields, enabling them to coordinate and optimize the overall performance of the controlled system at the system level.
