Robust and Adaptive Control With Aerospace Applications
Robust and Adaptive Control shows the reader how to produce consistent and accurate controllers that operate in the presence of uncertainties and unforeseen events. Driven by aerospace applications the focus of the book is primarily on continuous-dynamica
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Command Tracking and the Robust Servomechanism
3.1
Introduction
Most industrial control problems require the control system to accurately track commands. This requirement distinguishes these problems from regulation in which the state is driven to zero. From classical control theory, we know that in order to track a constant command with zero error, we need to add integral error control action into the controller. For single-input single-output (SISO) systems, the loop transfer function L(s) can be written as LðsÞ ¼
K ðb0 sm þ þ bm1 s þ 1Þ sp ða0 sn þ þ an1 s þ 1Þ
(3.1)
where the gain K and the polynomial coefficients ai and bi are real constants. The type of the control system depends upon the order p of the pole of L(s) at s ¼ 0. The number of finite zeros, their location, or the location of the poles are not important to specify the system type. The system type p, where p ¼ 0; 1; 2; indicates how many integrators are present in the control system. We know that in order to track a constant command rðtÞ ¼ constant , and to produce zero steady-state error, an integrator is needed, p 1, creating (at a minimum) a type 1 system. In order to track a type 1 input, the control system will need two integrators, creating a type 2 system. Thus, to track commands accurately, the class of commanded signals must be known, and the controller must be augmented with enough integrators to produce zero steady-state errors. When these integrators are added to the control system for command tracking, they also provide disturbance rejection within the same class, that is, a type 1 control system can track constant commands and reject constant disturbances. Similarly, a type 2 system can track ramp inputs and reject ramp disturbances. Basically, the augmentation of the system with these integrators for command tracking requires embedding into the system a model of the class of signals that the system will track. This is often referred to as the internal model principle [1]. E. Lavretsky and K.A. Wise, Robust and Adaptive Control, Advanced Textbooks in Control and Signal Processing, DOI 10.1007/978-1-4471-4396-3_3, # Springer-Verlag London 2013
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3 Command Tracking and the Robust Servomechanism
For instance, when tracking a constant command and adding a single integrator, we have embedded the command generation internal model r_ ¼ 0 into the system. In the previous chapter, we have illustrated the use of linear quadratic optimal control theory to design a controller and examined the excellent stability properties provided by that method. The linear quadratic regulator (LQR) forces the system state to go to zero, forming a type 0 control system. If one wants to track a constant command using such an LQR controller, the system would have a steady-state offset error to the command. We know from Eq. (3.1) that in order to track a constant command with zero error, we need to add an integrator, creating a type 1 control system. A natural extension of the LQR method presented in the previous chapter would b
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