Commencing with a nonlinear state space model, an equation is formed for the derivative of each output, equal in order to the rank of the plant with respect to the output concerned. Each of these equations has at least one control variable on the right hand side, together with state variables. For each of these equations, a corresponding differential equation of the same order and with the same output derivative on the left hand side is formed for the desired closed-loop dynamics. These equations are in terms of the output derivatives (which can be expressed in terms of the original plant state variables) and the reference inputs. Then the right hand sides of the plant output derivative equations are equated to the corresponding right hand sides of the desired closed-loop dynamic equations. The resulting set of simultaneous equations is solved for the control variables. The result is a nonlinear control law forcing the closed-loop system to obey the desired dynamics. The closed-loop system is also automatically decoupled. The basic control system is suitable for all minimum phase plants but requires modification for non-minimum phase plants, which would otherwise exhibit unstable zero dynamics.
The sytem robustness can be greatly improved by inserting a pure integrator before each reference input of the above control system and closing outer sliding mode control loops.
The method is demonstrated by simulation and by experiment for the speed control of electrical drives employing synchronous motors and induction motors. A particularly interesting feature of these control systems is that they have an internal, controlled oscillatory mode which automatically generates the three phase stator currents with continuously variable amplitude and frequency. There are various versions of the controller, one of which automatically ensures mutual orthogonality between the torque producing current and magnetic flux vectors, such as in the state-of-the art vector control methods.
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