Open-loop Instability Research Articles

The application of disturbance observer-based sliding mode control for magnetic levitation systems

A control algorithm for the position tracking of a magnetic levitation system is presented in this article. The magnetic levitation system is well known for its non-linear dynamic characteristics and open-loop instability. The external disturbances will deteriorate the dynamic performance of the magnetic levitation system, and may give rise to system instability. This problem triggers enormous interests in designing various controllers for the non-linear dynamic system. In this article, a magnetic levitation system is first modelled. Then, a sliding mode controller is proposed, with a simple yet effective disturbance observer to perform disturbance rejection. Both the simulation results and the experimental results verify the validity of the robust controller.

Multiscale Control for Nanoprecision Positioning Systems With Large Throughput

A problem of continuing interest in feedback control is handling conflicting time-domain performance specifications. Semiconductor manufacturing is one of the applications of particular interest in this context with the demanding feature sizes (on the order of a few tens of nanometers) to be produced on a wafer while still requiring high throughput (greater than 100 wafers per hour). In this brief, we propose a multiscale control design method based on a reduced-order model-following scheme for the dynamic systems with such conflicting time-domain performance requirements. This method uses a dynamic reference model to make the plant output track the model output as closely as possible without increasing the overall order of the control system. Optimal proportional-integral (PI) control is used, which is essentially a modification of the conventional optimal control. A detailed analytical proof is given to show that this control scheme effectively overcomes the limitations of the conventional optimal control techniques and provides consistent performances at nano- as well as macroscale positioning with fast rise and settling times. Benefits and limitations of the proposed control scheme are described and stability and performance analyses are discussed. A six-degree-of-freedom (6-DOF) extended-range magnetically levitated (maglev) nanopositioning stage, which is open-loop unstable, is used as a test bed to demonstrate the developed control strategy. Step responses under a variety of conditions are obtained to verify the effectiveness of the proposed method. This method exhibits significantly better and robust performances in terms of transient as well as steady-state behavior compared with conventional optimal-control schemes. Furthermore, it can be applied to a general class of higher-order linear time-invariant (LTI) systems with or without open-loop instability.

Output-feedback stabilization of an unstable wave equation

We consider the problem of stabilization of a one-dimensional wave equation that contains instability at its free end and control on the opposite end. In contrast to classical collocated “boundary damper” feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition, the controllers and the associated observers designed in the paper are more complex due to the open-loop instability of the plant. The controller and observer gains are designed using the method of “backstepping,” which results in explicit formulae for the gain functions. We prove exponential stability and the existence and uniqueness of classical solutions for the closed-loop system. We also derive the explicit compensators in frequency domain. The results are illustrated with simulations.

Electromagnetic Actuator Control: A Linear Parameter-Varying (LPV) Approach

This paper deals with system identification and control of a nonlinear electromagnetic actuator, which can be used in many practical applications: electromagnetic valve actuators of combustion engines, artificial heart actuators, magnetic levitation, electromagnetic brakes, etc. The considered practical control problem requires accurate control of the moving armature between two extreme positions. The main objective is to assure small contact velocity, which is known as "soft landing" of the moving armature, and, in this way, low-noise low-component-wear operation. First, due to open-loop instability, system-identification experiments are performed around different equilibrium positions under closed-loop control, and a linear parameter-varying (LPV) model and a bound of plant uncertainty are derived. Next, an LPV controller is designed in a robust control framework (robust gain-scheduled controller). Since the system evolves along quasi-equilibrium positions, quadratic and biquadratic analyses are performed using linear matrix inequalities. Finally, the experimental results show that the controller design problem can be handled successfully, considering an LPV approach. This paper reflects a pragmatic viewpoint: The control structure is simple and easy to implement, and offers good performance and robustness; therefore, it is suitable for industrial applications

Time-varying transfer function extraction of an unstable launch vehicle via closed-loop identification

The aerospace launch vehicle, developed today by manufacturers, is characterized by high nonlinearity, open loop instability and time-varying behaviors. The transfer functions of the vehicle can be extracted using well-known linearization methodology. This paper presents an alternative to obtain the transfer functions via closed-loop identification using six degrees of freedom nonlinear simulation software. The pitch program is taken into account as the external excitation. Control and stability of the process are performed using robust PID controller. The model structure with some unknown parameters is obtained after mathematical modeling, thus the case of our problem is a parameter identification one. Time-variant parameters are estimated by Kalman filter approach with the aid of ARX model structure.

Closing an open-loop control system: vestibular substitution through the tongue.

The human postural coordination mechanism is an example of a complex closed-loop control system based on multisensory integration [9,10,13,14]. In models of this process, sensory data from vestibular, visual, tactile and proprioceptive systems are integrated as linearly additive inputs that drive multiple sensory-motor loops to provide effective coordination of body movement, posture and alignment [5-8, 10, 11]. In the absence of normal vestibular (such as from a toxic drug reaction) and other inputs, unstable posture occurs. This instability may be the result of noise in a functionally open-loop control system [9]. Nonetheless, after sensory loss the brain can utilize tactile information from a sensory substitution system for functional compensation [1-4, 12]. Here we have demonstrated that head-body postural coordination can be restored by means of vestibular substitution using a head-mounted accelerometer and a brain-machine interface that employs a unique pattern of electrotactile stimulation on the tongue. Moreover, postural stability persists for a period of time after removing the vestibular substitution, after which the open-loop instability reappears.

Assessment of open-loop rollover control of articulated vehicles under different manoeuvres

In this paper, a comprehensive three-dimensional heavy vehicle model is developed to investigate the effectiveness of an open-loop roll instability control. The steering system compliance, roll steer, bump steer, Ackerman steer and wrap steer are incorporated in the vehicle model, along with a comprehensive tyre model and ABS algorithm. Time delays due to a driver's reaction and the transportation lag of the braking system are characterized by a variable called the reaction delay. The rollover indicators in terms of roll safety factor, tractor and trailer lateral accelerations and roll angles, and the rearmost axle roll angle are investigated for their effectiveness for open-loop roll stability control in various cornering and evasive manoeuvres, road conditions, braking efforts, and different reaction delays.

Multivariable gain-scheduled fuzzy logic control of an exothermic reactor

Fuzzy logic control frequently exhibits superior performance to classical linear controllers even for ‘hard’, mathematically well defined plants, as described in this paper. The case-study of a highly nonlinear exothermic continuous stirred tank reactor, which poses a multivariable control problem with two interacting loops and open-loop instability, is used. The behaviour of the fuzzy logic controller is compared with that of a PID controller. A smooth, easily tuneable gain-schedule is designed to handle offset-like problems with a fuzzy controller. It is analytically shown that such a gain-schedule is the simpler, intuitive equivalent of a manipulation of the corresponding fuzzy membership functions. The fuzzy controller structure chosen is a parsimonious one, with the choice of Gaussian bell-shaped membership functions generating a smooth input/output surface with nontrivial inferencing spanning the entire input space. This provides a clear, non-heuristic reason to select Gaussian over triangular shapes for membership functions. The gain-scheduled fuzzy controller shows excellent control performance, significantly outperforming the PID controllers in both servo and regulatory modes. The disturbance rejection behaviour of the modified fuzzy controller is observed to be particularly good.

Output regulation of nonisothermal plug-flow reactors with inlet perturbations

This work addresses the output regulation of flow systems described by a class of nonlinear partial differential equation (NPDE) systems with either one or two manipulated inputs. The main design procedures involves two steps, the first is to determine the feasible steady-state operation through the numerical computation, the second is to use the ‘boundary’ controls whose number and position can be directly planned. The developed control strategy can be treated as a typical linearization design, so that the outlet state of flow systems can be approximately linearized and maintain a certain degree of output tracking performance. Since the inlet perturbations may cause the hot spot phenomenon or open-loop instability of nonisothermal processes, the special two-point control scheme can suppress the effect of inlet disturbances and alleviate the control action. Finally, several issues are evaluated through simulations and implemented on an exothermic plug-flow reactor (PFR).

Steady-State and Dynamic Effects of Design Alternatives in Heat-Exchanger/Furnace/Reactor Processes

Feed-effluent heat exchangers (FEHE) are widely used in industry to preheat the feed to adiabatic tubular reactors. The hot reactor effluent is passed through a FEHE to recover heat. The positive feedback of energy introduces the potential for open-loop instability. Heat-exchanger bypassing is typically used to control the reactor inlet temperature. Previous papers1-4 have explored the control of this type of process. A furnace or heater following the FEHE may or may not be required under normal operation but is always needed for startup. Therefore, a design alternative exists in which both the reactor inlet temperature and the furnace inlet temperature are controlled, using the two manipulated variables: heat-exchanger bypassing and furnace firing rate. This paper explores the impact of these alternative designs on both the steady-state economics and the dynamic controllability. The exothermic, irreversible, gas-phase reaction A + B → C occurs in an adiabatic tubular reactor. A gas recycle returns uncon...

Effect of Kinetic, Design, and Operating Parameters on Reactor Gain

Many industrial reactions are carried out in adiabatic tubular reactors. The desired inlet feed temperature is usually achieved by using the hot reactor effluent to preheat the cold feed in a feed−effluent heat exchanger (FEHE). This positive feedback of energy introduces the potential for open loop instability. Previous papers have explored the control of this type of process. The key parameter in FEHE/reactor systems is the reactor gain, i.e. the change in the reactor exit temperature Tout for a given change in the reactor inlet temperature Tin: KR = ΔTout/ΔTin. The larger the reactor gain, the more likely open loop instability will occur and the more difficult the control problem will be. This paper explores the impact of various kinetic, design, and operating parameters on reactor gain. Three reaction systems are studied. The first is the hypothetical reaction A + B → C, where kinetic parameters (activation energy and heat of reaction) can be varied. Then two real industrial reactions are studied: the hydrodealkylation of toluene (HDA process) and the chlorination of propylene (allyl chloride process), where design parameters (inlet reactant feed concentration and reactor length) are varied. Larger reactor gains occur for increases in reactant feed concentration, activation energy, and heat of reaction. Reactor gains can vary non-monotonically for some design parameters when interactions between concentration and temperature effects occur.

Dynamics and Control of Membrane-Based Electrochemical Processes

Electrochemical reactors based on ionomeric membranes are the subject of much current industrial activity in a wide variety of applications, including fuel cells, salt splitting, electroorganic syntheses, metal recovery systems, ozone and hydrogen peroxide generation, and acid catalysis. These reactors are extremely interesting from a control perspective because they represent reaction/separation systems in which these two basic unit operations are intimately coupled, as in reactive distillation. In addition, there is evidence to suggest that gas-fed membrane electrolysis cells can exhibit open-loop instability at high current densities. Since process economics generally argue in favor of maximizing operating current densities (which determine production rate per unit cell area), there is a strong practical motivation for the development of stabilizing feedback controllers. As a first step toward this objective, this paper describes a first-principles model of the dynamics of a simple ionomeric membrane reactor system and some preliminary results concerning its qualitative behavior, particularly in the regime of high operating current densities.

Tuning Temperature Controllers on Openloop Unstable Reactors

There have been several methods proposed in the literature for tuning feedback controllers in openloop unstable processes. All of these methods assume a transfer function model is available that consists of a steady-state gain, deadtime, and positive pole. In some methods an additional negative pole is considered. This paper points out that this type of transfer function is often not appropriate for tuning temperature controllers in openloop unstable chemical reactors. There are two reasons for this. First, many chemical reactors used in industry have more than one positive pole. For example, the commonly used jacket-cooled continuous stirred-tank reactor has two positive poles even for the simple reaction A → B. Second, in practical applications we may not have a model, just an operating reactor. Even if a model is available, it is usually much more complex than the simple transfer function used in the literature. Openloop identification is not possible because the process is openloop unstable. Usually the only information available is the ultimate gain and the ultimate frequency, which have been obtained from a closedloop relay−feedback test. In this paper the effect of conversion on openloop instability is explored for jacket-cooled continuous stirred-tank reactor systems. Both linear and nonlinear models are used to study controller tuning. The Tyreus−Luyben (TL) tuning method is demonstrated to give more robust control for this type of system than the Ziegler−Nichols method. A procedure for obtaining an approximate gain/deadtime/positive-pole model is presented so that the tuning methods proposed in the literature can be used. However, these tuning methods provide performance that is no better than the TL method.

External versus Internal Open-Loop Unstable Processes

This paper points out that there are two distinctly different types of open-loop unstable processes and that this leads to different methods for the selection and tuning of feedback controllers. Internal open-loop instability occurs when an individual unit operation is open-loop unstable, i.e., one or more open-loop eigenvalues with positive real parts. The classical chemical engineering example is the exothermic irreversible chemical reactor. External open-loop instability occurs when a collection of open-loop stable units are connected in such a way that the coupled system is open-loop unstable. The classical chemical example is a feed/effluent heat exchanger connected with an adiabatic exothermic chemical reactor. These two types of processes have different dynamic structures. The transfer function of an internal open-loop unstable process is typically of order 2 or higher. The transfer function of an external open-loop unstable process typically has a net order of zero. This difference explains the phenomenon described by Tyreus and Luyben (J. Process Control 1993, 3 (4), 241−251) of using reset action to improve dynamic control, which conflicts with conventional wisdom that adding reset or integral action degrades dynamic performance.

Multiple Steady States and Instability in Distillation. Implications for Operation and Control

The fact that distillation columns, even in the ideal binary case, may display multiple steady states and unstable operating points has only recently been recognized. This article addresses some implications of these phenomena for the operation and control of distillation columns. Under manual operation, the multiplicity and instability will result in inability to reach separations corresponding to unstable operating points and may furthermore cause abrupt changes and hysteresis in operating conditions. It is shown that an unstable operating point may be stabilized by feedback control of a single product composition or tray temperature (onepoint control). The steady-state multiplicity does, in this case, not represent any severe limitation in operation, but if the control is not sufficiently tight, the column may settle in sustained oscillations (stable limit cycle). Finally, the impact of open-loop instability on the achievable closed-loop performance with both product compositions under feedback control is discussed.

Dynamic output feedback control of minimum-phase multivariable nonlinear processes

This paper concerns the synthesis of dynamic output feedback controllers for minimum-phase multivariable nonlinear processes with a nonsingular characteristic matrix. State-space controller realizations are derived that induce a linear input/output behavior of general form in the closed-loop system. A combination of input/output linearizing state feedback laws and state observers is employed for the derivation of the controllers. For open-loop stable processes, the process model is used as an open-loop state observer. In the more general case of possible open-loop instability, a reduced-order observer is used based on the forced zero dynamics of the process model. The performance and robustness characteristics of the proposed control methodology are illustrated through simulations in a chemical reactor example.

Unusual dynamics of a reactor/preheater process with deadtime, inverse response and openloop instability

This paper presents a mathematical analysis of the unusual dynamics found in a coupled reactor/preheater process. The outlet temperature of the reactor exhibits inverse response (wrong-way behaviour) for a change in the inlet reactor temperature and a large deadtime. The coupling of the preheater with the reactor produces positive feedback, which makes the coupled process openloop unstable. Both theoretical linear analysis and simulation studies reveal some unusual dynamics and some interesting frequency response plots. The Nyquist plot of the openloop characteristic equation of the reactor itself is used to determine the number of poles in the right half of the s-plane (RHP) of the openloop characteristic equation of the coupled reactor/preheater system. For small values of reactor gain and small values of the positive-zero time constant, there is one pole; so the closedloop system is stable if the total openloop system (process plus controller) Nyquist plot encircles negatively the (−1,0) point. As the gain and positive-zero time constant are increased, a point of discontinuity is reached. This occurs when the Nyquist plot of the openloop reactor indicates two poles instead of one in the RHP. Now there must be two negative encirclements of the (−1,0) point for closedloop stability. For small values of the positive-zero time constant and reactor gain, the Nyquist plot of the coupled system does show two encirclements. However, for a given deadtime, there are maximum values for the positive-zero time constant and the reactor gain beyond which the closeloop system cannot be stabilized. One of the most unusual dynamic features of this system is the effect that reset (integral action) in the controller has on closedloop stability. Conventional wisdom says that adding integral action in a feedback controller always degrades dynamic stability and performance. However, in this coupled reactor/preheater process the addition of integral action can improve closedloop stability.

Digital control of an electromagnetic suspension system using the TMS-32020 signal processor

Dynamics and control of electromagnetic suspension (EMS) systems for noncontacting vehicle suspension have been extensively studied during past two decades. Due to the inherent nonlinearities and open-loop instability of the EMS system, excitation control of vehicle suspension electromagnets is accomplished either through precision analogue controllers or by using large on-board computer systems. Drift and inflexibility are design constraints of the former, while the latter has significant capital costs. With the availability of low-cost, high-speed and wide data-byte VLSI devices, use of digital techniques to control high-speed analogue systems is now becoming feasible. This paper establishes a framework for the real-time control of this unstable nonlinear EMS system using the Texas Instruments TMS-32020 digital signal processor (DSP).

Stability of a PWM current-source rectifier/inverter fed induction motor drive

The stability of pulse width modulated current-source rectifier/inverter fed induction motor drives is considered in the paper. Rectifier/inverter operations are considerably complicated because of the high frequency pulse width modulated switching of power devices. To investigate the system stability successfully, an analysis without a complete simulation is performed by introducing synchronously rotating reference frames at the input and output terminals. The linearised equations are easily obtained so the intrinsic open-loop instability and the effects of circuit parameters under closed-loop operation can be made clear by determination of their dominant roots. The relevance of the results is verified by the transient characteristics of the nonlinear model, with and without a PI controller for constant V/f control.

Stability boundaries for aircraft with unstable lateral-directional dynamics and control saturation

Aircraft that do not possess inherent (aerodynamic) stability must rely on closed-loop control systems for stable operation. Because there are limits on the deflections of an aircraft's control surfaces, the region of stable operation also is bounded. These boundaries are investigated for a lateral-directional example in which vertical fin size is inadequate to provide directional stability and where aileron and rudder deflections are subject to saturation. Fourth-order models are used in this study, with flight control logic based on minimum-control-energy linear-quadratic-regulatory theory. It is found that the stability boundaries can be described by unstable limit cycles surrounding stable equilibrium points. Variations in regions of stability with gain levels and command inputs are illustrated. Current results suggest guidelines for permissible limits on the open-loop instability of an aircraft's lateral-directional modes.