Impulse Bit Research Articles

Side and front fast imaging of solid and liquid fed ablative pulsed plasma thruster’s discharge

Ultra-fast camera imaging is used to capture temporal evolution of the plasma discharge in an ablative Pulsed Plasma Thruster (PPT) fueled with solid polytetrafluoroethylene (PTFE) and liquid perfluoropolyether (PFPE), with the goal of comparing the mechanism of the expelled mass acceleration and getting insight into the physics of the processes involved in the discharge maintenance and impulse bit production. The fast photos are taken from two viewpoints (non-simultaneously), directing camera parallel (side view) and perpendicular (front view) to the ablative surface of the propellants, and are correlated with the relevant phases of the discharge current. Side view images reveal the presence of two distinct exhaust fractions, velocities of which differ by an order of magnitude. The speed of the faster fraction is comparable with the median of the ionized particles velocity expelled by the thruster. The vividness of the slow propagating component of the ablated mass, which mostly corresponds to neutral particles (based on velocity estimation) and lower mass bit of PFPE shots hint towards better propellant utilization by a liquid-fed PPT. Front view images confirm stable localization of PFPE discharge, forced by design of the propellant supplying plenum, whereas PTFE discharge is characterized by irregular path that covers an area of the propellant surface in random. The last half-period of the discharge is accompanied by vortices near the cathode spots. They are supposedly formed by evaporated electrode’s material and their trajectories imply further recombination in collisions with the electrodes, making this part of evaporated copper undetectable for charged particles diagnostics.

Analysis of composition and dynamics of the plasma plume emitted by a 1 J pulsed plasma thruster fed with polytetrafluoroethylene and determination of thruster efficiency components

The plasma plume of a 1 J pulsed plasma thruster fed with polytetrafluoroethylene (PTFE) was studied with electric probes to obtain the shape and composition of the beam of ejected ions. Two ion diagnostic tools—Faraday cup (FC) and retarding potential analyzer (RPA), were employed together with a time-of-flight approach. The FC was used to obtain spatially and time-resolved data of the mean ion charge expelled from the thruster in each pulse. With the RPA the beam was examined for the presence of specific ion species. The results of this investigation indicated the presence of both elements of PTFE in the beam—fluorine and carbon as well as copper from the discharge electrodes. Fluorine ions (identified in charge-states from F+ up to F6+) constituted the majority of ions in the plume with only trace amounts of C+ detected, which raises the question on the whereabouts of the remaining carbon. Energy distribution and relative abundance of fluorine ion species on axis were retrieved and it was found that F2+ constitutes over 40% of the plasma—in both quantity and energy fraction. Angular profiles of ion charge density, apart from the expected azimuthal asymmetry, showed heightened flux of ions in the area shaded by the discharge electrodes. The results obtained from both diagnostics allowed us to calculate propellant utilization, beam divergence, and energy utilization. By combining this information, the total thruster efficiency was retrieved, which turned out to be comparable to the value obtained from impulse bit measurements.

Design optimization of a low response time thruster control valve for small satellite missions

Small satellites and spacecrafts use propulsion systems to stabilize their attitude, control orientation and deorbit it at the end of the mission life. The flow and thrust of the liquid propulsion system are controlled by the control valve, typically operated by solenoid. According to recent literature, space-qualified solenoid valves are expensive and financially non viable, especially for small satellite missions. In this paper, a detailed methodology is presented to design the solenoid valve dedicated for small satellites. This study includes design criteria focusing on geometrical configuration. The dynamic, and electro-magnetic models are used to characterize the solenoid valve. Dynamic modelling helps to determine the response time of valve, affecting the thruster minimum impulse bit. The physical dimensions of the designed solenoid valve are smaller than those available in commercial off-the shelf (COTS) solenoid valves. However, the efficacy of the proposed design is proven effective with the COTS space grade valve specifications, and it shows that the proposed valve consumes less power than the conventional valves. Finally, it is worth noting that despite their modest and compact physical dimensions, the proposed valve are ideal for small satellite missions.

Implementation of direct battery driven discharge in triggerless operation of vacuum arc thrusters

The increasing demand for CubeSat missions with limited budgets and constrained timelines necessitates the development of innovative propulsion systems. This paper focuses on vacuum arc thrusters (VATs) and investigates the implementation of battery-driven discharge for triggerless firing. VATs offer a promising alternative for CubeSat propulsion, generating thrust through the pulsed ejection of plasma in a cathode arc discharge. The study explores the feasibility of direct battery-driven discharge in VATs and demonstrates stable and variable-duration pulses. Experimental testing using high C-rating lithium-ion (Li-ion) batteries showcase the advantages and operational benefits of this approach. The results highlight the potential for variable energy pulses, precise control of thrust impulse bit, and improved performance compared to capacitor-driven systems. Future research directions include optimising pulse duration for enhanced lifetime, plume analysis, and erosion rate investigations. The implementation of VATs with battery-driven discharge presents a promising solution to enhance the performance and reliability of CubeSat propulsion systems while ensuring compliance with space sustainability guidelines.

Experimental investigation on magnetic field strength providing thrust saturation in a magnetic nozzle radiofrequency plasma thruster

Magnetic field strength applied to a magnetic nozzle radiofrequency (rf) plasma thruster having a 10.5 cm diameter source tube is increased up to about 3 kG by pulsing the solenoid current. A target plate is installed at 30 cm downstream of the source and an impulse bit exerted to the target is measured to assess the thrust, where the thrust balance measurement was impossible due to the interaction between the pulsed magnetic fields and the eddy currents on surroundings. Since the diameter of the plasma plume at the target location is larger than the target diameter, a comparison between the thrust balance and target measurements under continuous magnetic field and rf power is performed prior to the pulsed magnetic field experiments, showing that about 65 percent of the plasma momentum is exerted to the target plate. Saturation of the impulse bit, being equivalent to the force multiplied by the rf pulse width, is clearly observed when increasing the magnetic field strength. The magnetic field providing the force saturation is found to be changed by the source diameter, which is qualitatively explained by considering a change in the plasma loss to the source wall in a thruster model containing the particle balance, power balance, and one-dimensional magnetic nozzle models. It is suggested that the magnetic field strength required for optimizing the force, i.e. the thrust, can be reduced when enlarging the source tube diameter.

Study on the influence of ignitor position on a coaxial pulsed plasma thruster

Because of their unique advantages, coaxial pulsed plasma thrusters are competitive propulsion systems for microsatellites. The ignitor of a coaxial pulsed plasma thruster can be mounted on the side or at the center, but previous studies have not compared the effects of the two mounting positions on the thruster. Therefore, this study designs and investigates a coaxial pulsed plasma thruster that permits the ignitor position to be changed while keeping the electrode geometry constant. The discharge circuit parameters, plume plasma characteristics, and propulsion performance are measured, and long-exposure photographs are taken. While the impulse bit of the thruster with a coaxially-mounted ignitor is slightly lower than with a side-mounted ignitor (319.5 μNs vs. 334.5 μNs), the ablated mass bit of the former is significantly lower (15.2 μg vs. 24 μg). The thruster with a coaxially-mounted ignitor achieves higher specific impulse and efficiency (2139 s and 16.74%). Because of the match between the electric field and the neutral gas supply during discharge in the thruster with a coaxially-mounted ignitor, its energy and propellant utilization is higher, thus achieving a better propulsion performance.

Design, Development, and Testing of a Low-Cost Sub-Joule µPPT for a Pocket-Cube

This paper presents the design and development of a sub-joule micro-Pulsed Plasma Thruster (µPPT) as a possible low-cost propulsion solution for Pocket-Cubes, used to increase its reliability, capability, and lifetime. It is shown that the µPPT successfully met Pocket-Cube design standards and additional requirements using the iterative design method, focused mainly on simplification and improvements to a traditional PPT design, the utilization of commercial-off-the-shelf (COTS) components and 3D printing. The µPPT was designed to operate and be controlled from an Arduino UNO with a main bank energy of 0.118 J to 0.272 J and power consumption of 0.5 W. It was successfully tested for performance and lifetime in a vacuum chamber (−720 mmHg to −96 kPa) with the use of a micro-pendulum test stand and a high-speed camera. The thruster was tested for its designed operation parameters of 3.3 V and 5 V at a pulsed frequency of 0.25/0.5 Hz. The test results showed that the optimal performance of the thruster with an input voltage supply of 5 V at a pulse frequency of 0.5 Hz, achieved a minimal impulse bit of 0.698 μNs and thrust range of 0.349~1.071 μN. A comparison to the STRaND-1 3U CubeSat’s PPT performance data showed that the developed µPPT is a competitive propulsion solution, as it achieved more thrust with a similar minimal impulse bit, using only one-third of the power consumption. During the lifetime testing, the µPPT was able to produce 1980 shots.

DESIGN AND PERFORMANCE OF A 110 N ATTITUDE CONTROL THRUSTER FOR LUNAR MISSIONS

The design and development of an attitude control thruster, designated A110, developed for broad attitude control applications, including lunar landers, is described. The A110 thruster produces 110 N (25 lbf) of nominal thrust and in-excess of 305.5 s of specific impulse (I<sub>sp</sub>) while operating on a fuel blend of 20% monomethylhydrazine-80% hydrazine (M20) with mixed oxides of nitrogen containing 3% nitric oxide (MON3) as the oxidizer. The injector and combustion chamber are additively manufactured with cobalt-chrome and niobium C103, respectively, to enable the rapid development required of the industry. The thruster is capable of response times of under 10 ms and currently produces a minimum impulse bit (MIB) of 0.436 N.s and has demonstrated operation in pulse-mode with a variety of duty cycles during development testing. These capabilities allow for the generation of precise pulses for accurate control of a lander. The testing of this thruster is used to evaluate steady-state and pulse-mode specific impulse, impulse repeatability, and combustion stability. The measured performance (i.e., specific impulse and characteristic velocity (C*), are compared with theoretical values. The capabilities demonstrated by the A110 thruster are applicable to a wide range of lunar lander missions and can be qualified for minor changes in thruster requirements for additional missions without the need for major design changes. This article is published with the permission of the authors granted to 3AF - Association Aéronautique et Astronautique de France (www.3AF.fr) organizer of the Space Propulsion International Conference.

Effect of absorption depth on chemical energy release from laser ablation of ADN-based liquid propellants

Ammonium dinitramide is a new green oxidant, which has been widely used in the field of liquid propulsion. In order to study the influence of doping laser absorber on the propulsion performance of liquid targets, the mixed solution of ammonium dinitramide and ionic liquid 1-allyl-3-methylimidazolium dicyandiamide was used as the ablation target, and the infrared dye was used as the laser absorber to change the doping content of infrared dye and laser energy density. The results show that the light absorption performance of liquid propellant is obviously improved with the increase of infrared dye content. The absorption coefficient and absorption depth of liquid propellant with infrared dye content of 0.6wt% are 425.19 cm−1 and 23.52 μm, respectively. The propellant with lower absorption depth produces high values of impulse bit, specific impulse, impulse coupling coefficient and ablation efficiency under the ablation of high laser energy density. The propellant sample doped with 0.6wt% infrared dye yielded the highest impulse bit of 114.32 μN·s, largest specific impulse of 84.14 s, highest impulse coupling coefficient of 1.07 mN·s·J−1, and the maximal ablation efficiency of 44% at the laser energy density of 21.51 J·cm−2. Furthermore, the effects of laser energy density and absorption depth on the chemical energy release of liquid propellant were studied. It was found that the maximum contribution of chemical energy to kinetic energy of ammonium dinitramide based liquid propellant is 93.93%. In general, improving the absorption performance of propellant can significantly enhance the energy release of energetic propellant. Which has guiding significance for propellant design and selection.

Performance of pulsed plasma thrusters with a non-volatile liquid propellant stored in porous ceramics with different pore diameters

Pulsed plasma thrusters (PPTs) typically have characteristics like a high specific impulse, a small volume, a low system mass when used in microsatellites. However, the lifetime of these thrusters is limited by their disadvantages, which include carbon deposition. A non-volatile liquid perfluoropolyether showed a good capability of lower carbon deposition in previous studies, but previous investigations focused on carbon deposition. The influence of the pore diameter of porous ceramics, which can change the supply rate, has not yet been studied. Perfluoropolyether, which is impregnated into the porous ceramics with different pore diameters (10 nm, 100 nm, 1 μm), was investigated in this study. A decreasing pore diameter of porous ceramics resulted in a lower resupply rate to reduce the ablated mass bit (the consumption of propellant per discharge) of PPTs. The use of perfluoropolyether resulted in an increase in the ablated mass bit and impulse bit, and a decrease in the specific impulse and efficiency. When using perfluoropolyether, the impulse bit increases with decreasing pore diameter of the porous ceramic. Perfluoropolyether achieved a higher specific impulse and efficiency with decreasing pore diameter of the porous ceramic, due to the increasing kinetic energy of plasma as a percentage of total energy.

Unsteady Stream-Tube Model for Pulse Performance of Bipropellant Thrusters

We propose a theoretical model for predicting the impulse bit of bipropellant thrusters under pulse-firing operations. The present theoretical model, which considers the nonuniformity of the mixture ratios created inside the thrust chamber, extends the stream-tube approach, which is limited to the prediction of the steady performance. In pulse-firing operation, the fuel or the oxidizer alone can be injected in isolation due to the mismatched injection timing before the rated injection, which leads to the deterioration of performance as compared with the steady operation. The present approach (the unsteady stream-tube model) successfully implements a time-dependent stream-tube structure inside the thrust chamber, allowing for the prediction of the impulse bit as a straightforward function of the injection conditions. Three different pulse-firing tests using distinct hypergolic propellants demonstrate the validity of this model, typically reproducing the notable trend of deterioration in the impulse bit in the short-pulsed mode. We also examine the time-averaged specific impulse and mass flow rate to improve the impulse bit during short-pulsed operations.

The influence of aluminum nanoparticles on the laser ablation characteristics of hydroxylamine nitrate-based liquid propellants

Hydroxylamine nitrate is a nontoxic green energetic propellant that has a high energy density and wide application prospects in the field of aerospace propulsion. Here, the laser ablation and propulsion properties of hydroxylamine nitrate-based propellants doped with aluminum nanoparticles were studied. The results showed a significant improvement in the laser absorption properties of hydroxylamine nitrate doped with aluminum nanoparticles. The maximum absorption rate increased by more than 80%, while the absorption depth decreased considerably with a minimum of 277 μm. The sputtering behavior also weakened during laser ablation, and the impulse of hydroxylamine nitrate could be enhanced by doping aluminum nanoparticles. With the 0.7wt% mass fraction of aluminum nanoparticles, a maximum impulse bit of 57.55 μN·s at 90 mJ laser energy. This is led to a maximum energy conversion efficiency of 218.1% and an enhancement of the kinetic energy of the hydroxylamine nitrate-based propellant of 56% with laser energy of 25 mJ. Compared with ordinary liquid propellants, doped samples have better propulsion performance.

Investigation on operational stability of a pulsed plasma thruster with a pressure probe

The Pulsed Plasma Thruster (PPT) was the first electric propulsion device to be implemented on a spacecraft in orbit (Zond 2) in 1964. However, the pulse ignition mode and the shot-to-shot discharge differences introduce challenges into the evaluation of the performance of PPTs. This work performs impact pressure measurements on a coaxial PPT (Pulsed Electric ThRuster of the University of Stuttgart, PETRUS) using a pressure probe. It is aimed to examine two specific thruster performance values that conventional thrust stands cannot easily obtain: the transient behavior during the phase where the final propellant surface contour is established (i.e., the shot-to-shot difference for the impulse bit during this phase) and the initial operation stability for the lifetime test. The impact pressure difference was measured at various discharge voltages using the pressure probe. To test the thruster's operational stability, a 5000 shots burn-in experiment was conducted on PETRUS. During the operation, the impulse bit variations were measured over 5000 continuous shots using the pressure probe. From the results, the impulse bit curve of PETRUS first shows a trend of decreasing impulse bit (the initial several hundred shots) and then an increasing trend (after 300 shots). After 2151 shots, the thruster's operating condition reaches a steady state with an impulse bit of 144 μN s. Additionally, the reliability of the pressure probe measurement is also evaluated by comparing the results with those from a thrust stand. The results are in good agreement at higher discharge voltages (from 0.9 kV to 1.6 kV), the maximum measured impulse bit difference of the pressure probe over the thrust stand is 9.7%. These results provide a dynamic evaluation of the thruster's operational stability from an engineering aspect. Correspondingly, the comparison results between the pressure probe and thrust stand is a preliminary verification of the pressure probe impulse bit measurement method in this work.

A Miniaturized, Green, End-Burning, and Sandwich Hybrid Propulsion System

Conventional hybrid rocket motors with thrust levels greater than 5 N rely on forced convection within the boundary layer as the primary heat transfer mechanism for fuel vaporization. For hybrid rockets with thrust levels of less than 5 N, oxidizer mass flow levels are sufficiently small that the rate of convective heat transfer is significantly reduced; and radiative heat transfer dominates the fuel vaporization mechanism. Radiative heating is a concern when implementing traditional hybrid rocket core-burn fuel grain designs for systems with thrust levels suitable for small satellites and CubeSats. Radiative heating induces a fuel-rich burn, which leads to low combustion efficiency and nozzle clogging. This paper presents a novel idea of using radiative heating in the design of a hybrid propulsion system suitable for CubeSats and small satellites. Additionally, this paper presents the test results of two fuel grain designs: an end-burning fuel grain design and a “sandwich” fuel grain design. This paper presents the test results of a 0.5 N end-burn system that produces a vacuum of 162 s with a rise time of 1.1 s and a minimum impulse bit of . This paper also presents the test results of a 1.0 N end-burn system that produces a vacuum of 136 s with a rise time of 2.2 s and a minimum impulse bit of . In addition, the test results for a 1 N sandwich system that produces a vacuum of 170 s with a rise time of 0.3 s and a minimum impulse bit of are presented. Acrylonitrile butadiene styrene, polyvinyl chloride, Nylon-12, and an acrylic plastic known as polymethyl methacrylate were used as fuel; and gaseous oxygen was used as the oxidizer during these testing campaigns. These propellants provide several advantages, including benign handling properties, simplified plumbing, and greater burn efficiency over traditional monopropellant hydrazine.

Effects of carbon, graphite, and graphene as propellant dopants in a laser-electric hybrid acceleration system

With the combination of laser ablation and electromagnetic acceleration, the laser–electric hybrid thruster is now perceived as an advanced propulsion system to support the precise maneuvering of microsatellites. To improve the performance of a laser–electric hybrid acceleration thruster, we used carbon, graphite, and graphene as dopants to modify the polymeric propellant fed into it. The impulse bit, consumed mass, discharge success rate, specific impulse, momentum coupling coefficient, and efficiency were measured and analyzed. Results showed that these three dopants led to higher values of impulse bit, specific impulse, discharge success rate, efficiency, and momentum coupling coefficient. The samples doped with carbon and graphene yielded better performances than those doped with graphite. Furthermore, the propellant sample made of 5% graphene and 95% polytetrafluoroethylene yielded the highest impulse bit of 394.7 μN·s, largest specific impulse of 1012.05 s, highest momentum coupling coefficient of 16.21 N/MW, and maximal efficiency of 8.21%. Furthermore, we compared the performances of the doped samples at different filling ratios and different operations. It was found that 5% was the optimal doping ratio for the dopants of carbon and graphene, and a higher discharge width could effectively improve the performances of the carbon-doped and graphene-doped samples.

A predictive model for macro-performances applied to laser-assisted pulsed plasma thrusters

The laser-assisted pulsed plasma thruster is considered a promising propulsion system to support the tasks of microsatellites because of its high specific impulse and low volume. Different from the traditional pulsed plasma thruster, laser-assisted pulsed plasma thruster uses the laser to replace the spark plug for ignition, which can avoid ignition failure and remove the side effect of carbon deposition. Both the thrust efficiency and impulse bit are expected to increase after the plasma flow produced by laser ablation is further ionized and accelerated. Since there are a few macro-performance prediction models in laser-assisted pulsed plasma thrusters, this paper develops a model based on the laser ablation model and electromagnetic acceleration model to capture macro-performances of laser-assisted pulsed plasma thrusters. In this model, the initial velocity and mass of plasma flow can be obtained from the ablation model, and the acceleration model is utilized to describe the electromagnetic acceleration process of plasma flow. With this combined model, the discharge current, voltage, impulse bit, specific impulse, and thrust efficiency can be estimated. The deviation between the predicted results and experimental results was less than 10%, verifying the correctness of the developed model. The effects of different parameters on the performance are further investigated with this model.

Dual mode operation of a hydromagnetic plasma thruster to achieve tunable thrust and specific impulse

We report here on initial studies of a pulsed hydromagnetic plasma gun that can operate in either a pre-filled or a gas-puff mode on demand. These modes enable agile and responsive performance through tunable thrust and specific impulse. Operation with a molecular nitrogen propellant is demonstrated to show that the hydromagnetic thruster is a candidate technology for air-harvesting and drag compensation in the very low Earth orbit. A dual mode operation is achieved by leveraging propellant gasdynamics to change the fill fraction and flow collisionality within the thruster. This results in the formation of distinct modes that are characterized by the current-driven hydromagnetic waves that they allow, namely, magneto-deflagration and magneto-detonation, respectively. These modes form the basis of using gasdynamics to enable responsive thruster performance. Using time-of-flight emission diagnostics to characterize near-field flow velocities, we find that a relatively dramatic transition occurs between modes as gas is allowed to expand in the thruster, with exhaust velocities ranging from 10 to 55 km/s in the deflagration and detonation regimes, respectively. Simulations of the processed mass bit offer the first glimpse into possible thruster performance and trade-offs between specific impulse and thrust. An impulse bit tunability of ∼22% is predicted, with differing propellant fill fractions when operating in a burst mode.

Influence of Different Energy Supply Methods on Performance of Ablative Pulsed Plasma Thrusters

For a certain configuration of ablative pulsed plasma thruster, changing the initial energy is an effective way to optimize the performance of the thruster. Different energy supply methods will not only affect the macro discharge characteristics and system performance of the thruster, but also affect the micro characteristics of the plasma in the discharge channel, the equivalent parameters of the discharge circuit, energy conversion efficiency and so on. In this study, the discharge characteristics, ablation characteristics and plasma motion characteristics of the thruster under different energy supply methods are analyzed experimentally and theoretically. The results show that using the method which includes increasing capacitance to heighten the initial energy of system under the same voltage, using a low-voltage, large-capacitor power supply method under the same initial energy, can effectively increase the impulse generated by Lorentz force and its proportion in the impulse bit. Moreover, the proportion of the ablative propellant effectively ionized and accelerated by Lorentz force increases. Therefore, the thruster has higher specific impulse and efficiency.

Impulse and electric charge characteristics of chemical propellant under pulsed laser irradiation

To enhance the thrust performance of laser-pulsed plasma thruster (LPPT), we proposed to use chemical propellant as substitute for ordinary PTFE propellant. The discharge characteristics and thrust performance of chemical propellant were analyzed. Meanwhile, the contribution of different energy sources and the physiochemical process between laser and chemical propellant were investigated. The results showed that the chemical propellant had higher plasma peak current and shorter discharge delay time compared to PTFE propellant. For chemical propellant, the contribution of laser energy was about 3%, while the electric energy and chemical energy was less than 17% and more than 80%, respectively. Based on the self-made torsion pendulum micro-impulse test bench, we found the chemical propellant had the highest impulse bit of 27.9 μN·s at 0 V and 69.9 μN·s at 1500 V. Meanwhile, the PTFE propellant had the lowest laser ablation mass, which is 3.70 μg/shot. The chemical reaction between laser and chemical propellant enhances the impulse bit, and increases the mass consumption of the propellant. The chemical propellant had the highest specific impulse, which is 295.1 s at 0 V and 738.5 s at 1500 V. These findings can indicate that the chemical propellant is a better choice than PTFE propellant.

Jet Propulsion Laboratory Small Satellite Dynamics Testbed Planar Air-Bearing Propulsion System Characterization

The Small Satellite Dynamics Testbed (SSDT) at the Jet Propulsion Laboratory offers infrastructure that supports small satellite guidance and control testing in a relevant dynamics environment while constrained by the gravity of Earth. One of the testing environments of the SSDT is its planar air-bearing platform, which has two lateral and one rotational degrees of freedom. This paper details the new position-control system and its compressed-gas thrusters of the planar air-bearing test platform, and characterizes the pertinent parameters related to the platform and propulsion system. A detailed description of the hardware and software systems of the platform is provided. Testing shows a maximum float time of 13.5 min, an average maximum thrust per thruster of 0.51 N, a tank pressure drop of 6.85 psi/s, and minimum impulse bit of 40 ms. The paper further presents results necessary for designing position controllers for use on this system, such as the relationship between number of thruster firing and output force, and the timing latency in the thruster firing control system. The paper shows that the embedded computer used in this platform exhibits timing delays of approximately 34 ms. Implications of these results on control system design are discussed in the conclusions.