3.0: Prelude

In Chapter 2, we discussed how to model a mechatronic system based on its physics. We also looked at both gray- and black-box approaches, where gray- and black-box approaches develop system models based on system input excitations and corresponding output responses. In this chapter, we will discuss the mechatronic system behaviors, including its stability and response performance indices. We will also address closed-loop control design and tuning of a mechatronic control system, including both model-free and model-based approaches.

Section 3.1 introduces these input-excitations used to study the performance of a mechatronic system, including impulse, step, and sinusoidal inputs. The system impulse response, in fact, is the inverse Laplace transform of the system transfer function. The step response of a mechatronic system can be associated with many performance indices such as system response time, settling time, system overshoot, and system steady state error. On the other hand, the system sinusoidal response provides its frequency response performance information [1], such as system magnitude and phase, as a function of frequency. This function provides key information for system characteristics such as phase and gain margins, system stability, and phase delay as a function of frequency.

Mechatronic system stability is a very important performance measure. Note that stability is an essential requirement for mechatronic system; if the system is not stable, the mechatronic system cannot respond to the command properly. In Section 3.2, we will define two kinds of mechatronic system stability characteristics for both linear and nonlinear mechatronic systems. These are asymptotical stability and bounded-input bounded-output (BIBO) stability. The asymptotical stability is associated with the system convergence property with any initial condition whereas BIBO stability is the characteristics of bounded system response with bounded input.

Many methods can be used to test system stability, including Lyapunov stability criteria for nonlinear and linear systems [2], closed-loop system pole locations, and system gain and phase margins for linear systems. For linear systems, the system transfer function pole locations can be used for not only stability analysis, but also for system performance study.

In Section 3.3, we analyze system response characteristics for the first and second order linear systems with associated system response time, settling time, and overshoot. For the first order systems, there is no response overshoot for the step input; we can calculate its settling time analytically using the system time constant. For the second order systems, we can calculate the system response characteristic analytically. The main difference between the first and second order systems is that there exists response overshoot for underdamped second order systems. We can approximate many high order system characteristics by those for either first or second order systems, which makes it important to become familiar with the system response characteristics of the first and second order systems. We will also analyze the relationship between system pole locations and response characteristics. These results can also be used to guide control design and tuning.

In practical applications, PID (proportional-integral-derivative) controllers are widely used due to their simplicity and easy gain tuning. In Section 3.4, we will discuss these three control parameters that make controller tuning easy. We will then address the Ziegler-Nichols PID gain tuning rule [3] in detail; normally the Ziegler-Nichols PID gains can be used as initial control gains for control gain tuning. We will also present an ad hoc (direct) PID control gain tuning method. In industrial applications, PID gains are often scheduled as a function of response error to improve closed-loop system performance. Examples will demonstrate how the gain-scheduled PID control can improve system performance.

Besides the PID control, model-based control techniques are often used for mechatronic systems. In Section 3.5, we will review the LQ (linear quadratic) and LQG (linear quadratic Gaussian) optimal control techniques [4], [5] for linear mechatronic systems. The main challenge of utilizing the LQ or LQG control for mechatronic systems is how to tune the model-based controller. Note that PID control is easy to tune because it has only three gains. We will also discuss the OCC (output covariance constraint) control technique [6], [7], [8] to address the issue of model-based control gain tuning because a family of OCC controllers can be generated with gains from low to high. To deal with nonlinear mechatronic systems, we can use the LPV (linear parameter-varying) control [9], [10], [11], [12], [13] to design gain scheduling controller for handling the mechatronic system nonlinearity.

In Section 3.6, we will review the machine-learning based scheduled PID control gain tuning. In Chapter 2, we discussed how to develop gray- and black-box models. Here, in Chapter 3, we discuss how to use machine learning to tune the gain-scheduled PID controller [15]. The gray- and black-box models can also be used for feedforward control [14]; however, they will not be addressed in this book.

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