Controllers are tuned either analytically or through optimisation techniques for integer, fractional and fractional complex order plants with dead time by considering only the positive frequency information of the plant. This is applicable only for integer / fractional order plants containing real coefficients which have an even symmetrical magnitude and odd symmetrical phase behaviour in frequency responses. The frequency response for (i) integer / fractional order plants containing complex coefficients and (ii) fractional complex order plants containing real / complex coefficients have unsymmetrical magnitude and phase behaviour. Tuning of controllers for such plants by considering only its positive frequency response produces reduced stability margins and in turn deteriorates its time response. Hence, tuning of controllers for such plants requires both positive and negative frequency information which in turn demands complex coefficient controllers. To address the problem, complex coefficient PI/PD/PID controllers are proposed in this paper. Unified expressions are derived for complex coefficient PI/PD/PID controllers by considering both positive and negative frequency information of plant with complex coefficients and complex order derivatives plus dead time. Numerical simulations are performed for three different case studies with the proposed complex coefficient integer order controllers and results are compared with real coefficient PI/PD/PID controllers under various operating conditions.

通过仅考虑设备的正频率信息，针对具有死区时间的整数、分数和分数复阶设备进行分析或通过优化技术调整控制器。这仅适用于包含实系数的整数/分数阶设备，这些实系数在频率响应中具有偶对称幅度和奇对称相位行为。(i) 包含复数系数的整数/分数阶设备和 (ii) 包含实数/复数系数的分数复数阶设备的频率响应具有不对称的幅度和相位行为。通过仅考虑其正频率响应来调整此类设备的控制器会降低稳定性裕度，进而降低其时间响应。因此，此类设备的控制器调整需要正负频率信息，而这又需要复系数控制器。为了解决这个问题，本文提出了复系数 PI/PD/PID 控制器。通过考虑具有复系数和复阶导数加上死区时间的对象的正负频率信息，推导出复系数 PI/PD/PID 控制器的统一表达式。使用所提出的复系数整数阶控制器对三个不同的案例研究进行了数值模拟，并将结果与​​各种操作条件下的实际系数 PI/PD/PID 控制器进行了比较。本文提出了复系数 PI/PD/PID 控制器。通过考虑具有复系数和复阶导数加上死区时间的对象的正负频率信息，推导出复系数 PI/PD/PID 控制器的统一表达式。使用所提出的复系数整数阶控制器对三个不同的案例研究进行了数值模拟，并将结果与​​各种操作条件下的实际系数 PI/PD/PID 控制器进行了比较。本文提出了复系数 PI/PD/PID 控制器。通过考虑具有复系数和复阶导数加上死区时间的对象的正负频率信息，推导出复系数 PI/PD/PID 控制器的统一表达式。使用所提出的复系数整数阶控制器对三个不同的案例研究进行了数值模拟，并将结果与​​各种操作条件下的实际系数 PI/PD/PID 控制器进行了比较。