Predictive Pressure Control of a Monopropellant Powered Actuator

This work presents a model-based predictive control methodology applicable to systems with discrete input values (on/off) subject to a pure time delay. Specifically, the goal of this work is to develop a controller that will track a desired pressure for a monopropellant-powered pneumatic system with binary on/off input valves. The pneumatic actuator is pressurized by the catalytic decomposition of the liquid monopropellant hydrogen peroxide. The challenge is tracking a proportional output with a binary input, which means the output amplitude is not proportional to the input amplitude, but is proportional in time. This requires the controller to predict the future states for each of the two input candidates, on or off. The error between the predicted future states and the current desired states is calculated for each of the inputs. A control candidate is chosen for the input valve based upon the decrement of a Lyapunov function. This results in the controller tracking a time delayed version of the desired pressure signal. Experimental results of the predictive controller demonstrate the effectiveness of this idea. INTRODUCTION Recent work by Goldfarb et al. [1] and Barth et al. [2] has presented an experimental monopropellant-powered actuation concept designed to address the energy and power density requirements imposed by a self-powered human-scale robot. One configuration of the above system controls the pressure in each side of a double-acting pneumatic piston via the direct injection of a monopropellant. This injection, resulting in pressurization, is controlled via a binary on/off solenoid valve, whereas depressurization is controlled with a proportional exhaust valve. The direct inject and subsequent catalytic decomposition of monopropellant used to pressurize the actuator presents several characteristics typically not seen in the control of more standard pneumatic actuators, namely: a discrete-valued control input (on or off), a non-negligible transport delay of the propellant in between the valve and the catalytic material, and the reaction dynamic of the chemical decomposition process within the motion control loop of the actuator. This work presents a model-based continuous predictive control methodology suitable for all three unique characteristics of the particular actuation concept being considered, as well as addressing more general systems with continuous plant dynamics, with or without a pure time delay, subject to a bang-bang type control input. Servo control of typical pneumatic systems is usually implemented using proportionally controlled servovalves. With this technique, pneumatic fluid flow is directed into or out of each piston chamber via proportional constriction through a valve orifice. This control method has been studied by many people [2-6]. An alternative to proportional servovalves in pneumatic control is the utilization of binary on/off solenoid valves. Binary valves eliminate the need to model the nonlinear flow characteristics through the proportional valve, and they are generally less expensive. Previous research has been conducted demonstrating the effectiveness of binary on/off valves utilized for the control of pneumatic actuators [7-14]. The work presented here draws from work addressing the control of typical pneumatic systems using on/off valves. In the method presented, the controller determines which of the two possible input candidates (valve closed or valve open) will better track a desired reference signal. One perspective of controlling a proportional output with a binary input is to view the control action as one in which proportionality in time, not amplitude, governs the continuous output. Real-time control precludes proportionality in time without a prediction method. The proposed controller predicts the states of the system at a specified future time for each of 1 Copyright © 2003 by ASME the input candidates. The error between the predicted states and the current desired states is calculated for each of the inputs and the input candidate that produces a decrement in a Lyapunov stability function is then chosen. This results in the controller tracking a time delayed version of the desired signal. The predictive control design method is experimentally implemented on the experimental monopropellant actuation system and results are presented. Predictive control has been a major research area for many years [15-21]. Predictive control uses a model of the system to predict future outputs based on the current available data. A predicted control input sequence is calculated so the error between the predicted output and desired future output is minimized. Only the first value in the predicted control sequence is applied, and the rest of the input sequence is discarded. These steps are repeated each sampling period [15]. One of the best-known model predictive controllers is generalized predictive control (GPC) formulated by Clarke et al. [17]. It is based upon a linear, discrete-time transfer function model and the control input sequence is determined by minimizing a quadratic cost function of the error between the predicted and the desired future output [16, 18]. GPC analytically solves for the open-loop optimal control sequence given a continuous and unbounded control input, and current system states. Given a discrete-valued control input (i.e. bang-bang), the analytical solution for the future control sequence is not available and must be searched exhaustively from a large number of input sequence permutations. The control methodology proposed in this paper is based upon a continuous-time model whereby a subset of all possible future control input functions are evaluated based on their predicted response, and the control input candidate that drives the error toward zero is selected Work done by Demircioglu and Karasu [15] presents a continuous time version of GPC. However, their system does not use a binary on/off input. Tsang and Clarke [20] studied bang-bang GPC, where the control signal can take one of two values. Since the control can have two values, the input vector u can have 2 combinations, where NU is a future time instant called the control horizon. They propose using an exhaustive search of these combinations to find the control input that minimizes the GPC cost function. This may not be practical for the real time application considered here, so the proposed controller examines two input candidates and chooses the one that results in the smallest error. Sprock and Rau [21] formulate the predictive control for switch-mode applications where the on/off states of the input switches are chosen each sampling interval such that the difference between the reference state and the predicted state is minimized. Their system is modeled as a discrete-time state equation without a time delay, while the system described by this paper uses a continuous-time state equation with time delay. PREDICTIVE CONTROLLER DESIGN Consider a continuous, linear time-invariant state space model with time delay Tu: ) ( B ) ( Ax ) ( x u T t u t t − + = & (1) Consider the response of the system TU+TD seconds in the future, where TD is the prediction horizon. The response of the system TU+TD seconds into the future is given in closed form as: τ − τ + = + + ∫ + + τ − + + + d T u e t e T T t u T T t t T T t T T D U D U D U D U ) ( B ) ( x ) ( x̂ ) ( A ) ( A (2) For a binary-valued input u (i.e. “on” or “off”), consider two candidate input functions from time t-TU to time t+TD (since u(t+TD) will affect states up to ), ) ( x̂ D U T T t + +      = τ = + τ − τ + = + +