Linear Time-invariant State Feedback Research Articles

An Explicit Parametrization of Closed Loops for Spatially Distributed Controllers With Sparsity Constraints

In this article,we study the linear time-invariant state-feedback controller design problem for distributed systems. We follow the recently developed system level synthesis (SLS) approach and impose locality structure on the resulting closed-loop mappings; the corresponding controller implementation inherits this prescribed structure. In contrast to existing SLS results, we derive an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">explicit</i> (rather than implicit) parameterization of all achievable stabilized closed-loops. This admits more efficient IIR representations of the temporal part of the closed-loop dynamics, and it allows for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_2$</tex-math></inline-formula> design problem with closed-loop spatial sparsity constraints to be converted to a standard model matching problem, with the number of transfer function parameters scaling linearly with the closed-loop spatial extent constraint. We illustrate our results with two applications: consensus of first-order subsystems and the vehicular platoons problem. In the case of first-order consensus, we provide analytic solutions and further analyze the architecture of the resulting controller implementation. Results for infinite extent spatially invariant systems are presented to provide insight to the case of a large but finite number of subsystems.

Exponential Stabilization of Linear Time-Varying Differential Equations with Uncertain Coefficients by Linear Stationary Feedback

We consider a control system defined by a linear time-varying differential equation of n-th order with uncertain bounded coefficients. The problem of exponential stabilization of the system with an arbitrary given decay rate by linear static state or output feedback with constant gain coefficients is studied. We prove that every system is exponentially stabilizable with any pregiven decay rate by linear time-invariant static state feedback. The proof is based on the Levin’s theorem on sufficient conditions for absolute non-oscillatory stability of solutions to a linear differential equation. We obtain sufficient conditions of exponential stabilization with any pregiven decay rate for a linear differential equation with uncertain bounded coefficients by linear time-invariant static output feedback. Illustrative examples are considered.

Regional robust stabilisation and domain-of-attraction estimation for MIMO uncertain nonlinear systems with input saturation

ABSTRACTThe regional robust stabilisation by means of linear time-invariant state feedback control for a class of uncertain MIMO nonlinear systems with parametric uncertainties and control input saturation is investigated. The nonlinear systems are described in a differential algebraic representation and the regional stability is handled considering the largest ellipsoidal domain-of-attraction (DOA) inside a given polytopic region in the state space. A novel set of sufficient Linear Matrix Inequality (LMI) conditions with new auxiliary decision variables are developed aiming to design less conservative linear state feedback controllers with corresponding larger DOAs, by considering the polytopic description of the saturated inputs. A few examples are presented showing favourable comparisons with recently published similar control design methodologies.

A linear-quadratic coordination control procedure with event-based communication

This paper deals with infinite-horizon quadratic optimal control for coordinated linear systems, a particular class of hierarchical control systems with a top-to-bottom information structure. In contrast to the case of monolithic linear systems, the corresponding optimal control law is not a linear time-invariant state feedback. We derive an approximation to the optimal control law, combining a linear state feedback with event-based bottom-to-top communication. The performance of this control procedure is illustrated by simulations.

Stabilization and Disturbance Attenuation Over a Gaussian Communication Channel

We consider the problem of stabilizing an unstable system driven by a Gaussian disturbance using a feedback signal transmitted over a memoryless Gaussian communication channel. By applying the concept of entropy power, we show that the mean square norm of the state vector must satisfy a lower bound that holds for any causal, measurable communication and control strategies that result in signals having well defined differential entropy. In addition, we show that use of nonlinear, time varying strategies does not allow stabilization over a channel with a lower signal-to-noise ratio than that achievable with linear time invariant state feedback. Finally, we show that for scalar systems the lower bound on the mean square norm of the state is tight, and achievable using linear time invariant communication and control.

A unified method for the design of nonovershooting linear multivariable state-feedback tracking controllers

We consider the use of linear multivariable feedback control to achieve a nonovershooting step response. A method is given for designing a linear time invariant state feedback controller to asymptotically track a constant step reference with zero overshoot and arbitrarily small rise time, under some mild assumptions. We present a unified design method that can be applied to continuous and discrete time systems, square and non-square systems, minimum and nonminimum phase systems, and also strictly proper and nonstrictly proper systems.

Optimal gain-scheduled flight control system design using a new fuzzy clustering algorithm

This article presents an analytical framework for the design of autopilots using fuzzy systems. A new fuzzy clustering method is presented in this article, which is used to model a non-linear flight vehicle using a set of linear models. It is shown that the proposed fuzzy clustering technique produces lower estimation errors compared with other fuzzy clustering techniques; it means that the modelling error using the technique introduced is lower than other clustering methods. The membership functions and rule sets, which are obtained by fuzzy clustering, are then applied to a set of linear time-invariant optimal state-feedback controllers, obtained for each rule, towards extraction of the global non-linear controller matching closely with the dynamic properties and changes in the plant. The stability of the fuzzy model and the fuzzy system is established by the Lyapunov-based linear matrix inequality analysis. Simulation studies are reported to demonstrate the merits of the fuzzy set-based modelling and control approach in handling the demanding non-linear modelling and control task.

Invariant Manifold Approach for the Stabilization of Nonholonomic Chained Systems: Application to a Mobile Robot

In this paper it is shown that a class of n-dimensional nonholonomic chained systems can be stabilized using the invariant manifold approach. First, we derive an invariant manifold for this class of systems and we show that, once on it, all the closed-loop trajectories tend to the origin under a linear smooth time-invariant state feedback. Thereafter, it is shown that this manifold can be made attractive by means of a discontinuous time- invariant state feedback. Finally, a mobile robot is taken as an example demonstrating the effectiveness of our study.

Memoryless H∞ controller for singular systems with delayed state and control

In this paper, the memoryless H ∞ controllers for singular systems with delayed state and control are presented. Based on the modified Riccati-equation approach, a memoryless linear time-invariant state feedback control law is developed. The controller, which is a delay-independent stabilizer for the singular delayed system will simultaneously reduce the H ∞ norm of the closed-loop transfer function from the disturbance to the controlled output to a prescribed level. An illustrative example is included to demonstrate our proposed approach.

Robust state feedback control of LTV systems: nonlinear is better than linear

Examples of linear control systems with fast time-varying uncertain coefficients are given, which can be stabilized by a nonlinear memoryless state feedback, but cannot be stabilized by a linear time-invariant dynamic state feedback. By means of one of these examples the authors show that the closed loop quadratic stability margin may be infinitely smaller than the actual stability margin.

Robust H∞ controller design for time-varying uncertain linear systems with time-varying state and control delays

The robust H∞ control problem for time-varying uncertain linear systems with time-varying state, and control delays is considered. Based on the Riccati equation approach, we design a robust H∞ linear time-invariant state feedback control law, which stabilizes the system and reduces the effect of the disturbance input on the controlled output to a prescribed level for all admissible uncertainties. A sufficient condition for the existence of the. robust H∞ controller is derived and two synthesis methods are presented for a robust H∞ control law to uncertain time-delay systems. One is to construct a robust H∞ control law by an equivalent linear time-invariant structural description for the sufficient condition and the control law can be obtained by solving an algebraic Riccati equation (ARE). The other method is to transform the sufficient condition into a linear matrix inequality (LMF) problem and the control law can be obtained by solving the LMI. A numerical example is presented to illustrate the obtained results.

Robust memoryless H∞ controller design for linear time-delay systems with norm-bounded time-varying uncertainty

We consider the robust H∞ control problem for linear time-delay systems with norm-bounded time-varying uncertainty. Based on the Riccati-equation approach, we present a method for designing a robust memoryless linear time-invariant state feedback control law, which stabilizes the system and reduces the effect of the disturbance input on the controlled output to a prescribed level for all admissible uncertainties.

Some new results on closed-loop stability in the presence of control saturation

Some new sufficient conditions for determining global asymptotic stability of an equilibrium point under control saturation are presented. For an open-loop stable system, we show that there exists a class of linear time invariant state feedback matrices which globally stablizes an equilibrium under control saturation. In addition, we show that for open-loop unstable plants, an equilibrium can never be globally stabilized using a certain class of saturated linear time invariant state feedback. Finally, we extend the stability results to nonlinear systems whose nonlinearities are globally Lipschitz.

Memoryless H∞ controller design for linear systems with delayed state and control

We present a memoryless H∞ control design method for linear time-invariant systems that have no parametric uncertainty but delay in the state and control. Based on the Riccati-equation approach, we design a memoryless linear time-invariant state feedback control law, which guarantees the asymptotic stability of the closed-loop control system and reduces the effect of the disturbance input on the controlled output to a prescribed level.

A contraction property for state feedback design of linear discrete-time systems

The contraction property is proposed as a tool for design of linear discrete-time control systems. Explicit formulae are given for all linear time-invariant state feedback controllers that imply the contraction property. The obtained controllers are tuned with only two user parameters. The two parameters are closely connected to the settling time and the damping of the closed loop system, respectively. The approach is illustrated by means of simulations.

Observer-based control of uncertain systems with non-linear uncertainties

Global asymptotic stabilization of nominally linear uncertain systems is considered where uncertain elements in the plant are modelled as cone bounded non-linearities. At first, a linear time-invariant state feedback law ensuring global asymptotic closed-loop stability is obtained. The algorithm which calculates such a feedback law is simple and straightforward. It does not involve repeated solutions of a parameter-dependent algebraic Riccati or any such non-linear equations. Any state feedback law thus developed can then be implemented via an observer specially designed to preserve the global asymptotic stability of the closed-loop system.

Stabilization of Uncertain Systems with Norm Bounded Uncertainty—A Control Lyapunov Function Approach

A robust stabilization problem in a state-space setting is treated. It is assumed that the states are available for feedback. Using a fixed Lyapunov function approach (quadratic stability) it is shown that an open loop stabilizability condition is equivalent to the existence of a stabilizing memoryless linear state-feedback controller. As a consequence, it is shown that the existence of a quadratically stabilizing nonlinear time-varying dynamic state-feedback controller implies the existence of a quadratically stabilizing memoryless linear time-invariant state-feedback compensator.

A design technique for the incomplete state feedback problem in multivariable control systems

The problem of choosing the “best” state variables to measure from a set of measurable state variables in a linear, time-invariant multivariable system with linear time-invariant incomplete state feedback is considered. An extension of Rosenbrock's Modal Analysis Technique is used. Essentially, the method is as follows: A measurement matrix C' consisting of l dominant eigenvectors of the transpose of the system matrix A is found, implying that all states of the system are measured. An approximate measurement matrix is then found so that those states which contribute negligibly to C'x and so that those states which cannot be physically measured are eliminated. This means that only the more significant states which contribute to the measurement of the l dominant modes of the system are retained. Experimental results, should be obtained by computer simulation, are given for two high-order systems—a Boiler System (9th order) and a Nuclear Reactor (11th order).