Dynamics of Linear Systems with Semiactive Force Generators

A semiactive element produces control forces in a dynamic system by controlled variation of a resistive parameter. Control constraints exist making the output of a semiactive element complex and nonlinear even for linear feedback control laws. State equations are derived for a general linear system with an arbitrary number of semiactive elements. These equations are piecewise-linear and exhibit intermittent linear dependencies among elements of the state vector. Two special cases exist yielding simplified descriptions of the semiactive control and system dynamics. Semiactive vibration control is applied to a flexible vehicle on a rough guideway. Numerical results demonstrate performance superior to passive suspensions and comparable to fully-active control. PPLICATION of modern control theory and parameter optimization techniques to shock and vibration control permit the design of high-performance suspension systems for taxiing aircraft1 and ground vehicles.2"7 Due to their sim- plicity, low cost, and reliability, passive suspensions com- posed of spring and damper elements are used in all but the most critical applications. Although control using active force generators receives considerable attention in the literature, its superior performance over passive control is frequently offset in practical designs by increased cost, complexity, energy requirements, and weight. While passive devices are self- powered and self-contained, an active force generator requires a power source to generate control inputs along with measurement and signal-processing systems. At least the disadvantages of a power supply can be eliminated by parametric control in which the parameters of passive elements are varied in response to a control signal. When a resistive parameter is varied, large control forces in a dynamic system may be generated by controlled dissipation of system energy. This type of control, which has been called semiac- tive, 8 combines features of passive control with the capability of generating the sophisticated feedback controls usually reserved for active systems. Recently, semiactive control has been studied in shock and vibration applications.8"11 In linear and nonlinear systems with deterministic8"10 and random inputs,9"11 these studies have established that the performance of semiactive force generators approaches that of fully active elements. This is not too surprising because any stable vibration-control system must eventually dissipate system vibrational energy, something a resistor does quite well. Although these studies illustrate basic applications and principles of semiactive control, a general treatment of the dynamics of a system with an arbitrary number of semiactive elements is lacking. Semiactive control introduces complex, nonstationary dynamics in even simple, linear systems.8'9 Because control forces are produced by a dissipative process, an inequality constraint on control power exists causing complex behavior of the semiactive control force. In this paper, a general theory of the dynamics of systems with semiactive elements is developed. Two special cases are found that provide sim- plified representations of the semiactive control and system dynamics.