1.3 Finite-Time and Finite-Frequency
9
the existing frequency domain research results, most of the results are based on the
full frequency band, but it rarely refers to the finite frequency domain.
In practical engineering, except white noise, many signals including disturbance
[226, 227] or reference input signals [228, 229], are evenly distributed in the finite
frequency domain, for example, the frequency range of the energy of seismic wave
signal is mostly concentrated in 0.3–8 Hz [230], the frequency range of the ground
vibration caused by high-speed train is mainly concentrated in 0–100 Hz [231],
the frequency range that human body is more sensitive to external force is 4–8 Hz
[232, 233], etc. Therefore, the system will obtain better performance in the design
of control system, if the controller can be designed based on the signal and the finite
frequency domain. In general, due to different frequency ranges, it is required to meet
different performance indexes in different frequency range so as to get the optimal
system performance. For example, for digital filters, in low frequency range, the
controllers are needed to have high gain which can ensure strong anti-interference
ability of the system; while in high frequency range, the low gain is needed to reduce
the influence of model uncertainty on system performance [234]. It means that the
frequency performance should be fully considered when we analyze and synthesize
the system. In conclusion, for multi-model jumping system which is more close to
actual engineering, the finite frequency performance study can not only improves
the theoretical system, but also solves the engineering application problems.
In general, we have three main methods to study the finite frequency control,
i.e., classical control method, frequency weighting method and General Kalman–
Yakubovich–Popov (GKYP) lemma method.
For classical control method, we introduce additional zeros and poles of controllers to make the controlled system have the desired frequency response performance. This method is only applicable to the single-input and single-output linear
time invariant system. It can not deal with multi-input and multi-output systems, highorder systems and other complex systems. Moreover, in classical control method, it
usually applies the frequency performance curve to analyze the transient and steadystate performance in time domain. It does not incorporate the finite frequency information into the controller design, and it is difficult to deal with the actual finite
frequency performance research.
Frequency weighting method bases on state space equation. It overcomes the
shortcoming of classical control methods which are not suitable for complex systems.
In frequency weighting method, we transform the finite frequency domain problem
of the original system into the full frequency domain problem of the composite
system by designing appropriate weighted transfer functions. This method not only
directly applies the control theory to the finite frequency control, but also reflects
the importance of each component of multi-dimensional signal in the system. In
[235], a frequency weighting method for order reduction of equilibrium models
was proposed for H 2 and H ∞ mixed sensitivity problems. For other conclusions
on frequency weighting method, we can refer to [236–239]. However, frequency
weighting method also has many disadvantages. Firstly, it is difficult to find a suitable
weighting function, and the process is very complex and is highly dependent on the
experience. Secondly, the performance quantitative information in finite frequency
9
the existing frequency domain research results, most of the results are based on the
full frequency band, but it rarely refers to the finite frequency domain.
In practical engineering, except white noise, many signals including disturbance
[226, 227] or reference input signals [228, 229], are evenly distributed in the finite
frequency domain, for example, the frequency range of the energy of seismic wave
signal is mostly concentrated in 0.3–8 Hz [230], the frequency range of the ground
vibration caused by high-speed train is mainly concentrated in 0–100 Hz [231],
the frequency range that human body is more sensitive to external force is 4–8 Hz
[232, 233], etc. Therefore, the system will obtain better performance in the design
of control system, if the controller can be designed based on the signal and the finite
frequency domain. In general, due to different frequency ranges, it is required to meet
different performance indexes in different frequency range so as to get the optimal
system performance. For example, for digital filters, in low frequency range, the
controllers are needed to have high gain which can ensure strong anti-interference
ability of the system; while in high frequency range, the low gain is needed to reduce
the influence of model uncertainty on system performance [234]. It means that the
frequency performance should be fully considered when we analyze and synthesize
the system. In conclusion, for multi-model jumping system which is more close to
actual engineering, the finite frequency performance study can not only improves
the theoretical system, but also solves the engineering application problems.
In general, we have three main methods to study the finite frequency control,
i.e., classical control method, frequency weighting method and General Kalman–
Yakubovich–Popov (GKYP) lemma method.
For classical control method, we introduce additional zeros and poles of controllers to make the controlled system have the desired frequency response performance. This method is only applicable to the single-input and single-output linear
time invariant system. It can not deal with multi-input and multi-output systems, highorder systems and other complex systems. Moreover, in classical control method, it
usually applies the frequency performance curve to analyze the transient and steadystate performance in time domain. It does not incorporate the finite frequency information into the controller design, and it is difficult to deal with the actual finite
frequency performance research.
Frequency weighting method bases on state space equation. It overcomes the
shortcoming of classical control methods which are not suitable for complex systems.
In frequency weighting method, we transform the finite frequency domain problem
of the original system into the full frequency domain problem of the composite
system by designing appropriate weighted transfer functions. This method not only
directly applies the control theory to the finite frequency control, but also reflects
the importance of each component of multi-dimensional signal in the system. In
[235], a frequency weighting method for order reduction of equilibrium models
was proposed for H 2 and H ∞ mixed sensitivity problems. For other conclusions
on frequency weighting method, we can refer to [236–239]. However, frequency
weighting method also has many disadvantages. Firstly, it is difficult to find a suitable
weighting function, and the process is very complex and is highly dependent on the
experience. Secondly, the performance quantitative information in finite frequency