H∞ surge control in centrifugal compressors by using active magnetic bearings (AMBs) has been suc... more H∞ surge control in centrifugal compressors by using active magnetic bearings (AMBs) has been successfully designed and implemented. However, the structure and design process of H∞ surge control are quite complex. This paper reports on the design and implementation of fractional-order proportional-derivative control (FOPD) that results in the required specifications of surge control with a simple controller structure. To validate its effectiveness, the proposed FOPD surge controller has been implemented on a centrifugal compressor test rig equipped with AMBs. Simulation and experimental results show that the FOPD surge controller outperforms integer-order PD (IOPD) control in extending the surge limit in terms of the mass flow and provides similar performance as the H∞ controller.
One of the key issues in control design for Active Magnetic Bearing (AMB) systems is the tradeoff... more One of the key issues in control design for Active Magnetic Bearing (AMB) systems is the tradeoff between the simplicity of the controller structure and the performance of the closed-loop system. To achieve this tradeoff, this paper proposes the design of a fractional order Proportional-Integral-Derivative (FOPID) controller. The FOPID controller consists of only two additional parameters in comparison with a conventional PID controller. The feasibility of FOPID for AMB systems is investigated for rotor suspension in both the radial and axial directions. Tuning methods are developed based on the evolutionary algorithms for searching the optimal values of the controller parameters. The resulting FOPID controllers are then tested and compared with a conventional PID controller, as well as with some advanced controllers such as Linear Quadratic Gausian (LQG) and H ∞ controllers. The comparison is made in terms of various stability and robustness specifications, as well as the dimensions of the controllers as implemented. Lastly, to validate the proposed method, experimental testing is carried out on a single-stage centrifugal compressor test rig equipped with magnetic bearings. The results show that, with a proper selection of gains and fractional orders, the performance of the resulting FOPID is similar to those of the advanced controllers.
This paper deals with consensus control of a class of nonlinear systems with Lipschitz nonlineari... more This paper deals with consensus control of a class of nonlinear systems with Lipschitz nonlinearities. Certain features of the Laplacian matrix are explored in the real Jordan form to identify conditions for global consensus control. Under the identified conditions, consensus control and stability of the proposed control are analysed in time-domain through Lyapunov functions in real domain. The proposed control uses relative state information of the system. A simulation study is included to demonstrate the proposed control designs with some simulation results shown.
This paper examines the asymptotic stabilizability of discrete-time linear systems with delayed i... more This paper examines the asymptotic stabilizability of discrete-time linear systems with delayed input. By explicit construction of stabilizing feedback laws, it is shown that a stabilizable and detectable linear system with an arbitrarily large delay in the input can be asymptotically stabilized by either linear state or output feedback as long as the open loop system is not exponentially unstable (i.e., all the open loop poles are on or inside the unit circle.) It is further shown that such a system, when subject to actuator saturation, is semi-globally asymptotically stabilizable by linear state or output feedback.
International Journal of Robust and Nonlinear Control, 2004
Stabilization problems of linear systems with right invertible constraints have mostly been solve... more Stabilization problems of linear systems with right invertible constraints have mostly been solved for both global and semi-global frameworks. Linear systems subject to non-right invertible constraints have completely different characteristics compared to those with right invertible constraints. This paper addresses the control problems of linear systems subject to non-right invertible constraints. This paper also expands further the available taxonomy of constraints. The new categorizations of constraints introduced reveal the difficulty in semi-global oriented control design for the non-right invertible constraints. Some new notions of stabilization are also introduced.
International Journal of Robust and Nonlinear Control, 1997
DOI to the publisher's website. • The final author version and the galley proof are versions of t... more DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:
International Journal of Robust and Nonlinear Control, 2009
In this paper, we study the problem of disturbance attenuation by output feedback for linear syst... more In this paper, we study the problem of disturbance attenuation by output feedback for linear systems subject to actuator saturation. A nonlinear output feedback, expressed in the form of a quasi-linear parametervarying system with state-dependent scheduling parameter, is constructed that leads to the attenuation of the effect of the disturbance on the output of the system. The level of disturbance attenuation is measured in terms of the restricted L 2 gain and the restricted L 2-L ∞ gain over a class of bounded disturbances.
IEE Proceedings - Control Theory and Applications, 1999
The uncertain systems under considered depend on norm-bounded time-varying matrices of uncertain ... more The uncertain systems under considered depend on norm-bounded time-varying matrices of uncertain parameters. The authors develop a quadratic guaranteed cost control method for robust stabilisation via linear memoryless state feedback and static output feedback. Also presented is a robust state feedback controller and static output feedback controller. It is shown that the state feedback controller and output feedback controller can be constructed by solving a linear matrix inequality.
The successful industrial application of flexible rotors supported on active magnetic bearings (A... more The successful industrial application of flexible rotors supported on active magnetic bearings (AMBs) requires careful attention not only to rotordynamic design aspects, but also to electromagnetic and feedback control design aspects. This paper describes the design, construction and modeling process for an AMB test rig which contains a 1.23m long flexible steel rotor, with a mass of 44.9 kg and two gyroscopic disks. The rotor typifies a small industrial centrifugal compressor designed to operate above 12,000 rpm and the first bending natural frequency. There are four AMBs-two AMBs at the shaft ends to support the shaft with a combined load capacity of 2600N and two additional AMBs at the mid and quarter spans to allow for the application of simulated destabilizing fluid or electromagnetic forces to the rotor. Simulated aerodynamic cross coupling stiffness values are to be applied to the rotor through these two internal AMBs with the goal of developing stabilizing robust controllers. The unique design allows multiple support and disturbance locations providing the ability to represent a variety of machine configurations, e.g., between bearing and overhung designs. The shaft transfer function in lateral movement has been developed with finite element model and then verified by experimental frequency response measurements. Models for the power amplifiers, position sensors, signal conditioning and data converter hardware were developed, verified experimentally and included in the overall system model. A PID controller was developed and tuned to levitate the rotor and enable further system characterization.
IEEE Transactions on Intelligent Transportation Systems, 2014
This paper studies the flocking problem for a multiagent system, where each agent is a vehicle wi... more This paper studies the flocking problem for a multiagent system, where each agent is a vehicle with nonholonomic dynamics. In particular, we consider the case where the agents are subjected to an arbitrarily large communication delay. A distributed low gain control law is derived based on the gradient of an artificial potential function. We demonstrate using the Lyapunov functional approach that the proposed control law drives the multiagent system into the stable flocking behavior. The effectiveness of the proposed control law is verified in numerical simulation.
The paper presents a method for designing output feedback laws that stabilize a linear system sub... more The paper presents a method for designing output feedback laws that stabilize a linear system subject to actuator saturation with a large domain of attraction. This method applies to general linear systems including strictly unstable ones, and is presented in both continuous-time and discrete-time setting. A nonlinear output feedback controller is first expressed in the form of a quasi-LPV system. Conditions under which the closed-loop system is locally asymptotically stable are then established in terms of the coefficient matrices of the controller. The design of the controller (coefficient matrices) that achieves a large domain of attraction is then formulated and solved as an optimization problem with LMI constraints.
This technical note proposes a switching anti-windup design, which aims to enlarge the domain of ... more This technical note proposes a switching anti-windup design, which aims to enlarge the domain of attraction of the closed-loop system. Multiple anti-windup gains along with an index function that orchestrates the switching among these anti-windup gains are designed based on the min function of multiple quadratic Lyapunov functions. In comparison with the design of a single anti-windup gain which maximizes a contractively invariant level set of a single quadratic Lyapunov function as a way to enlarge the domain of attraction, the use of multiple Lyapunov functions and switching in the proposed design allows the union of the level sets of the multiple Lyapunov functions, each of which is not necessarily contractively invariant, to be contractively invariant and within the domain of attraction. As a result, the resulting domain of attraction is expected to be significantly larger than the one resulting from a single anti-windup gain and a single Lyapunov function. Indeed, simulation results demonstrate such a significant improvement.
It was recently established that a second-order antistable linear system can be semiglobally stab... more It was recently established that a second-order antistable linear system can be semiglobally stabilized on its null controllable region by saturated linear feedback and a higher order linear system with two or more antistable poles can be semiglobally stabilized on its null controllable region by more general bounded feedback laws. We will show in this note that a system with three real-valued antistable poles cannot be semiglobally stabilized on its null controllable region by the simple saturated linear feedback.
We present controller design methods to smooth the discontinuity resulting from a piecewise linea... more We present controller design methods to smooth the discontinuity resulting from a piecewise linear control law which was proposed to improve the convergence performance for systems with input constraints. The continuous control laws designed in this note are explicit functions of the state and are easily implementable. We also show that the convergence performance can be further improved by using a saturated high-gain feedback law. The efficiency of the proposed methods is illustrated with the PUMA 560 robot model.
Low-gain feedback has found several applications in constrained control, robust control, and nonl... more Low-gain feedback has found several applications in constrained control, robust control, and nonlinear control. In this paper, we first generalize the existing low-gain design methods by introducing the notion of-vanishment and by providing a full characterization of feedback gains that achieve such a property. We observe that low-gain feedback can lead to energy peaking, namely, the control energy required by low-gain feedback increases toward infinity as the low-gain parameter decreases to zero. Motivated by this observation, we consider the notion of 2-vanishment and establish several of its characterizations, based on which a new design approach referred to as the 2 low-gain feedback approach for linear systems is developed. Different from the low-gain feedback, the 2 low-gain feedback is instrumental in the control of systems with control energy constraints. As an application of 2 low-gain feedback, the problem of semiglobal stabilization of linear systems with control energy constraints is solved in this paper. The notions of and 2-vanishment also allow us to establish a systematic approach to the design of and 2 low-gain feedback. The advantage of this new design approach is that it results in a family of control laws, including those resulting from the existing design methods. Index Terms-Constrained control, energy constraints, and 2 low-gain feedback, and 2-vanishment, parametric Lyapunov equation, semiglobal stabilization, slow peaking. I. INTRODUCTION L OW-GAIN feedback was initially proposed in [18] to deal with semiglobal stabilization of linear systems under actuator saturation. A key feature of the low-gain feedback, parameterized in a low-gain parameter, is that if the linear system is asymptotically null controllable with bounded controls (ANCBC), then for any given bounded initial condition,
IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 2002
This paper is devoted to stability analysis of linear systems with state delay and input saturati... more This paper is devoted to stability analysis of linear systems with state delay and input saturation. The domain of attraction resulting from an a priori designed state feedback law is analyzed using Lyapunov-Razumikhin and Lyapunov-Krasovskii functional approach. Both delay-independent and delay-dependent estimation of the domain of attraction are presented using the linear matrix inequality technique. The problem of designing linear state feedback laws such that the domain of attraction is enlarged is formulated and solved as an optimization problem with LMI constraints. Numerical examples are used to demonstrate the effectiveness of the proposed design technique.
International Journal of Robust and Nonlinear Control, 2009
The problem of robust global stabilization of linear systems subject to input saturation and inpu... more The problem of robust global stabilization of linear systems subject to input saturation and input-additive uncertainties is revisited in this paper. By taking advantages of the recently developed parametric Lyapunov equation-based low gain feedback design method and an existing dynamic gain scheduling technique, a new gain scheduling controller is proposed to solve the problem. In comparison with the existing H 2-type gain scheduling controller, which requires the online solution of a state-dependent nonlinear optimization problem and a state-dependent H 2 algebraic Riccati equation (ARE), all the parameters in the proposed controller are determined a priori. In the absence of the input-additive uncertainties, the proposed controller also partially recovers Teel's H ∞-type scheduling approach by solving the problem of global stabilization of linear systems with actuator saturation. The H ∞-type scheduling approach achieves robustness not only with non-inputadditive uncertainties but also requires the closed-form solution to an H ∞ ARE. Thus, the proposed scheduling method also addresses the implementation issues of the H ∞-type scheduling approach in the absence of non-input-additive uncertainties.
H∞ surge control in centrifugal compressors by using active magnetic bearings (AMBs) has been suc... more H∞ surge control in centrifugal compressors by using active magnetic bearings (AMBs) has been successfully designed and implemented. However, the structure and design process of H∞ surge control are quite complex. This paper reports on the design and implementation of fractional-order proportional-derivative control (FOPD) that results in the required specifications of surge control with a simple controller structure. To validate its effectiveness, the proposed FOPD surge controller has been implemented on a centrifugal compressor test rig equipped with AMBs. Simulation and experimental results show that the FOPD surge controller outperforms integer-order PD (IOPD) control in extending the surge limit in terms of the mass flow and provides similar performance as the H∞ controller.
One of the key issues in control design for Active Magnetic Bearing (AMB) systems is the tradeoff... more One of the key issues in control design for Active Magnetic Bearing (AMB) systems is the tradeoff between the simplicity of the controller structure and the performance of the closed-loop system. To achieve this tradeoff, this paper proposes the design of a fractional order Proportional-Integral-Derivative (FOPID) controller. The FOPID controller consists of only two additional parameters in comparison with a conventional PID controller. The feasibility of FOPID for AMB systems is investigated for rotor suspension in both the radial and axial directions. Tuning methods are developed based on the evolutionary algorithms for searching the optimal values of the controller parameters. The resulting FOPID controllers are then tested and compared with a conventional PID controller, as well as with some advanced controllers such as Linear Quadratic Gausian (LQG) and H ∞ controllers. The comparison is made in terms of various stability and robustness specifications, as well as the dimensions of the controllers as implemented. Lastly, to validate the proposed method, experimental testing is carried out on a single-stage centrifugal compressor test rig equipped with magnetic bearings. The results show that, with a proper selection of gains and fractional orders, the performance of the resulting FOPID is similar to those of the advanced controllers.
This paper deals with consensus control of a class of nonlinear systems with Lipschitz nonlineari... more This paper deals with consensus control of a class of nonlinear systems with Lipschitz nonlinearities. Certain features of the Laplacian matrix are explored in the real Jordan form to identify conditions for global consensus control. Under the identified conditions, consensus control and stability of the proposed control are analysed in time-domain through Lyapunov functions in real domain. The proposed control uses relative state information of the system. A simulation study is included to demonstrate the proposed control designs with some simulation results shown.
This paper examines the asymptotic stabilizability of discrete-time linear systems with delayed i... more This paper examines the asymptotic stabilizability of discrete-time linear systems with delayed input. By explicit construction of stabilizing feedback laws, it is shown that a stabilizable and detectable linear system with an arbitrarily large delay in the input can be asymptotically stabilized by either linear state or output feedback as long as the open loop system is not exponentially unstable (i.e., all the open loop poles are on or inside the unit circle.) It is further shown that such a system, when subject to actuator saturation, is semi-globally asymptotically stabilizable by linear state or output feedback.
International Journal of Robust and Nonlinear Control, 2004
Stabilization problems of linear systems with right invertible constraints have mostly been solve... more Stabilization problems of linear systems with right invertible constraints have mostly been solved for both global and semi-global frameworks. Linear systems subject to non-right invertible constraints have completely different characteristics compared to those with right invertible constraints. This paper addresses the control problems of linear systems subject to non-right invertible constraints. This paper also expands further the available taxonomy of constraints. The new categorizations of constraints introduced reveal the difficulty in semi-global oriented control design for the non-right invertible constraints. Some new notions of stabilization are also introduced.
International Journal of Robust and Nonlinear Control, 1997
DOI to the publisher's website. • The final author version and the galley proof are versions of t... more DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:
International Journal of Robust and Nonlinear Control, 2009
In this paper, we study the problem of disturbance attenuation by output feedback for linear syst... more In this paper, we study the problem of disturbance attenuation by output feedback for linear systems subject to actuator saturation. A nonlinear output feedback, expressed in the form of a quasi-linear parametervarying system with state-dependent scheduling parameter, is constructed that leads to the attenuation of the effect of the disturbance on the output of the system. The level of disturbance attenuation is measured in terms of the restricted L 2 gain and the restricted L 2-L ∞ gain over a class of bounded disturbances.
IEE Proceedings - Control Theory and Applications, 1999
The uncertain systems under considered depend on norm-bounded time-varying matrices of uncertain ... more The uncertain systems under considered depend on norm-bounded time-varying matrices of uncertain parameters. The authors develop a quadratic guaranteed cost control method for robust stabilisation via linear memoryless state feedback and static output feedback. Also presented is a robust state feedback controller and static output feedback controller. It is shown that the state feedback controller and output feedback controller can be constructed by solving a linear matrix inequality.
The successful industrial application of flexible rotors supported on active magnetic bearings (A... more The successful industrial application of flexible rotors supported on active magnetic bearings (AMBs) requires careful attention not only to rotordynamic design aspects, but also to electromagnetic and feedback control design aspects. This paper describes the design, construction and modeling process for an AMB test rig which contains a 1.23m long flexible steel rotor, with a mass of 44.9 kg and two gyroscopic disks. The rotor typifies a small industrial centrifugal compressor designed to operate above 12,000 rpm and the first bending natural frequency. There are four AMBs-two AMBs at the shaft ends to support the shaft with a combined load capacity of 2600N and two additional AMBs at the mid and quarter spans to allow for the application of simulated destabilizing fluid or electromagnetic forces to the rotor. Simulated aerodynamic cross coupling stiffness values are to be applied to the rotor through these two internal AMBs with the goal of developing stabilizing robust controllers. The unique design allows multiple support and disturbance locations providing the ability to represent a variety of machine configurations, e.g., between bearing and overhung designs. The shaft transfer function in lateral movement has been developed with finite element model and then verified by experimental frequency response measurements. Models for the power amplifiers, position sensors, signal conditioning and data converter hardware were developed, verified experimentally and included in the overall system model. A PID controller was developed and tuned to levitate the rotor and enable further system characterization.
IEEE Transactions on Intelligent Transportation Systems, 2014
This paper studies the flocking problem for a multiagent system, where each agent is a vehicle wi... more This paper studies the flocking problem for a multiagent system, where each agent is a vehicle with nonholonomic dynamics. In particular, we consider the case where the agents are subjected to an arbitrarily large communication delay. A distributed low gain control law is derived based on the gradient of an artificial potential function. We demonstrate using the Lyapunov functional approach that the proposed control law drives the multiagent system into the stable flocking behavior. The effectiveness of the proposed control law is verified in numerical simulation.
The paper presents a method for designing output feedback laws that stabilize a linear system sub... more The paper presents a method for designing output feedback laws that stabilize a linear system subject to actuator saturation with a large domain of attraction. This method applies to general linear systems including strictly unstable ones, and is presented in both continuous-time and discrete-time setting. A nonlinear output feedback controller is first expressed in the form of a quasi-LPV system. Conditions under which the closed-loop system is locally asymptotically stable are then established in terms of the coefficient matrices of the controller. The design of the controller (coefficient matrices) that achieves a large domain of attraction is then formulated and solved as an optimization problem with LMI constraints.
This technical note proposes a switching anti-windup design, which aims to enlarge the domain of ... more This technical note proposes a switching anti-windup design, which aims to enlarge the domain of attraction of the closed-loop system. Multiple anti-windup gains along with an index function that orchestrates the switching among these anti-windup gains are designed based on the min function of multiple quadratic Lyapunov functions. In comparison with the design of a single anti-windup gain which maximizes a contractively invariant level set of a single quadratic Lyapunov function as a way to enlarge the domain of attraction, the use of multiple Lyapunov functions and switching in the proposed design allows the union of the level sets of the multiple Lyapunov functions, each of which is not necessarily contractively invariant, to be contractively invariant and within the domain of attraction. As a result, the resulting domain of attraction is expected to be significantly larger than the one resulting from a single anti-windup gain and a single Lyapunov function. Indeed, simulation results demonstrate such a significant improvement.
It was recently established that a second-order antistable linear system can be semiglobally stab... more It was recently established that a second-order antistable linear system can be semiglobally stabilized on its null controllable region by saturated linear feedback and a higher order linear system with two or more antistable poles can be semiglobally stabilized on its null controllable region by more general bounded feedback laws. We will show in this note that a system with three real-valued antistable poles cannot be semiglobally stabilized on its null controllable region by the simple saturated linear feedback.
We present controller design methods to smooth the discontinuity resulting from a piecewise linea... more We present controller design methods to smooth the discontinuity resulting from a piecewise linear control law which was proposed to improve the convergence performance for systems with input constraints. The continuous control laws designed in this note are explicit functions of the state and are easily implementable. We also show that the convergence performance can be further improved by using a saturated high-gain feedback law. The efficiency of the proposed methods is illustrated with the PUMA 560 robot model.
Low-gain feedback has found several applications in constrained control, robust control, and nonl... more Low-gain feedback has found several applications in constrained control, robust control, and nonlinear control. In this paper, we first generalize the existing low-gain design methods by introducing the notion of-vanishment and by providing a full characterization of feedback gains that achieve such a property. We observe that low-gain feedback can lead to energy peaking, namely, the control energy required by low-gain feedback increases toward infinity as the low-gain parameter decreases to zero. Motivated by this observation, we consider the notion of 2-vanishment and establish several of its characterizations, based on which a new design approach referred to as the 2 low-gain feedback approach for linear systems is developed. Different from the low-gain feedback, the 2 low-gain feedback is instrumental in the control of systems with control energy constraints. As an application of 2 low-gain feedback, the problem of semiglobal stabilization of linear systems with control energy constraints is solved in this paper. The notions of and 2-vanishment also allow us to establish a systematic approach to the design of and 2 low-gain feedback. The advantage of this new design approach is that it results in a family of control laws, including those resulting from the existing design methods. Index Terms-Constrained control, energy constraints, and 2 low-gain feedback, and 2-vanishment, parametric Lyapunov equation, semiglobal stabilization, slow peaking. I. INTRODUCTION L OW-GAIN feedback was initially proposed in [18] to deal with semiglobal stabilization of linear systems under actuator saturation. A key feature of the low-gain feedback, parameterized in a low-gain parameter, is that if the linear system is asymptotically null controllable with bounded controls (ANCBC), then for any given bounded initial condition,
IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 2002
This paper is devoted to stability analysis of linear systems with state delay and input saturati... more This paper is devoted to stability analysis of linear systems with state delay and input saturation. The domain of attraction resulting from an a priori designed state feedback law is analyzed using Lyapunov-Razumikhin and Lyapunov-Krasovskii functional approach. Both delay-independent and delay-dependent estimation of the domain of attraction are presented using the linear matrix inequality technique. The problem of designing linear state feedback laws such that the domain of attraction is enlarged is formulated and solved as an optimization problem with LMI constraints. Numerical examples are used to demonstrate the effectiveness of the proposed design technique.
International Journal of Robust and Nonlinear Control, 2009
The problem of robust global stabilization of linear systems subject to input saturation and inpu... more The problem of robust global stabilization of linear systems subject to input saturation and input-additive uncertainties is revisited in this paper. By taking advantages of the recently developed parametric Lyapunov equation-based low gain feedback design method and an existing dynamic gain scheduling technique, a new gain scheduling controller is proposed to solve the problem. In comparison with the existing H 2-type gain scheduling controller, which requires the online solution of a state-dependent nonlinear optimization problem and a state-dependent H 2 algebraic Riccati equation (ARE), all the parameters in the proposed controller are determined a priori. In the absence of the input-additive uncertainties, the proposed controller also partially recovers Teel's H ∞-type scheduling approach by solving the problem of global stabilization of linear systems with actuator saturation. The H ∞-type scheduling approach achieves robustness not only with non-inputadditive uncertainties but also requires the closed-form solution to an H ∞ ARE. Thus, the proposed scheduling method also addresses the implementation issues of the H ∞-type scheduling approach in the absence of non-input-additive uncertainties.
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