Synthesis Of Linear Control Systems Research Articles

Synthesis of Linear Control Systems with Maximum Speed and Given Overshoot

W e st udy the solution of the so-called maximum speed problem of a linear control system, in which, unlike the classical optimal speed problem with a relay-type control, a linear control algorithm is determined for a linear object that provides the maximum speed of the system. It is of practical importance due to the widespread practical application of linear co ntrol laws. The problem is formulated in relation to continuous, one-dimensional high-order control objects described by the corresponding transfer function or an equivalent system of ordinary differential equations in a certain phase space. The time of the transition process t tp of the designed system is understood in the sense of the classical theory of automatic control and is determined using the zone D = s pr 4,321 %, equal to the given (desired) value of the overshoot of the synthe si zed s ystem. This overshoot corresponds to an oscillatory link with a damping coefficient √ 2/2 — the Butterworth filter of the second order. Accordingly, the maximum performance problem is put in the following formulation: it is required to fi nd a linear feedback algorithm that provides the closed control system with a given astatism order n a and transferring the control object from the initial state to the final one, determined by the constant signal of the regulator’s task, with the minimum value of transient time t tp and set value overshoot s = s pr when performing a constraint on the control signal |u(t)| m u max . At present, this problem has been solved by an approximately algebraic method of synthesizing linear control systems in determining the desired transfer function of a projected closed system based on typical (reference) normalized transfer functions (NTF). The approximate nature of the decision is determined by the fact that the NTF used in the synthesis of high-speed systems are established empirically. This paper proposes a mathematically sound solution to the problem of maximum speed using the theory of analytical design of optimal controllers (ADOC). The maximum speed and set limits on the overshoot and the value of the control signal in the synthesized system are provided by the proposed method for selecting the weights of the quadratic quality functional. We emphasize that the proposed method for the synthesis of high-speed systems, in contrast to the algebraic method, is applicable to a wider class of control objects: both minimal-phase and non-minimal-phase, as containing zeroes, and not. The method is illustrated by an example of the synthesis of a high-speed fourth-order control system containing the results of its simulation.

Stability analysis and control synthesis for linear systems with non-symmetrical input saturation

This paper studies the stability and control problem of linear systems with non-symmetrical input saturation. A system with non-symmetrical input saturation is transformed into a system with switching symmetrical input saturation. A switching controller is designed based on a parametric algebra Riccati equation, dwell time and the equivalent switched system. Exponential stability is guaranteed with the proposed switching controller. The main advantages of the proposed method lie in reducing the conservatism caused by directly using symmetrical input saturation control and increasing the state convergent speed. The designed controller can be computed easily by solving the Riccati equation. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

The Solution of a Problem of Speed of Response on Output Coordinate for Linear Dynamic Systems

The solution of the so-called problem of speed of response in one coordinate, which has important theoretical and practical importance, is investigated. It is formulated with reference to linear one-dimensional high-order control objects described by a system of ordinary differential equations in a certain phase space. The transient time tnn of the system designed is understood in a sense of the classical control theory in reference to one (output) coordinate of the object and is determined by using the zone Δ = σ* = 4.321 %, which equals the given (desirable) value of the overshoot of the system synthesized. This overshoot corresponds with the speed of response oscillating second-order element with a damping coefficient ζ= = 2 2 0,7071 / . It is indispensable to mention here that the equation Δ = σ is one of the necessary conditions for the maximum speed of response of the system with the oscillating character of transient processes. In accordance to this the task of the speed of response by one coordinate can be described by the following generalized formulation: one must find the linear algorithm of the feedback signal, which provides a preset order of the astatism na for the closed-loop control system and converts the control object from a zero state into a final state, which is determined by the constant signal of the input, with a minimal time value of the transient processes of the system tnn and the preset value of the overshoot σ m σ* while fulfilling the constraint of the control signal |u(t)| m umax. Nowadays the task mentioned is approximately solved by the algebraic method of the synthesis of linear control systems with the determination of a desirable transfer function of the designed closed-loop system based on model normalized transfer functions (NTF). In the works by Kim D. P. there was carried out the analysis of four types of normalized transfer functions characterized by the increased speed of response. In this work two additional types of normalized transfer functions are suggested, in comparison with mentioned NTF they have the increased speed of response in case of the preset value of the overregulation σ* = 4.321 %. On their basis and using the methodology of the modal control the method of the synthesis of the controller is suggested; this method ensures the transient time of the designed system to be close to the minimum in case of the preset constraint of the overregulation and the value of the control signal. It needs to be emphasized that in contrast to the algebraic method of the synthesis, this method is applied to a wider range of control objects: as to minimal-phased objects as to non-minimum-phased ones; as to the objects containing zeros as to those without them. The method is illustrated by an example of synthesis of control system speed of response of the fourth order, containing the results of its modeling.

Robust Control Policies Given Formal Specifications in Uncertain Environments

We consider robust control synthesis for linear systems with complex specifications that are affected by uncertain disturbances. This letter is motivated by autonomous systems interacting with partially known, time-varying environments. Given a specification in bounded linear temporal logic, we propose a method to synthesize control policies that fulfill the specification for all considered disturbances, where the disturbances affect the atomic propositions of the specification. Our approach relies on introducing affine disturbance feedback policies and casting the problem as a bilinear optimization problem. We introduce an inner approximation of the constraint set leading to a mixed-integer quadratic program that can be solved using general-purpose solvers. The framework is demonstrated on a numerical case study with applications to autonomous driving.

Language-Guided Controller Synthesis for Linear Systems

This paper considers the problem of controlling discrete-time linear systems from specifications given as formulas of syntactically co-safe linear temporal logic over linear predicates in the state variables. A systematic procedure is developed for the automatic computation of sets of initial states and feedback controllers such that all the resulting trajectories of the closed-loop system satisfy the given specifications. The procedure is based on the iterative construction and refinement of an automaton that enforces the satisfaction of the formula. Linear programming based approaches are proposed to compute the polytope-to-polytope controllers that label the transitions of the automaton. Extensions to discrete-time piecewise affine systems and specifications given as formulas of full linear temporal logic are included. The algorithms developed in this paper were implemented as a software package that is available for download. Their application and effectiveness are demonstrated for several case studies.

グラフ電卓を利用した制御系解析・設計用ライブラリ

The graphing calculator, Voyage200, is a calculator which has the Derive system as a computational engine and can do symbolic manipulation. It has a large liquid crystal display (128 × 240 pixels) as for a calculator and various responses can be drawn on it. Furthermore, it can be easy to make user-defined functions on it and extend to solve specified problems. Taking notice of the high potential and its portability, we develop the library for analysis and synthesis of linear control systems. In this paper, we introduce typical programs including our library which consists of programs more than 100 and examine the effectiveness at the stand point of control education. According to the results of several examples, our system may be available for the 4th order systems or less, as regards computational time. However, it must be enough as education of the beginners of control engineering and powerful handheld tool.

Control Synthesis for Linear Systems by Methods of Stability Theory of Motion

Control Synthesis for Linear Systems by Methods of Stability Theory of Motion

Improved stability analysis and gain-scheduled controller synthesis for parameter-dependent systems

We present new algorithms for robust stability analysis and gain-scheduled controller synthesis for linear systems affected by time-varying parametric uncertainties. These new techniques can also be applied to parameter-dependent nonlinear systems with real-rational nonlinearities. Sufficient conditions for robust stability, as well as conditions for the existence of a robustly stabilizing gain-scheduled controller, are given in terms of a finite number of linear matrix inequalities (LMIs); explicit formulas for constructing robustly stabilizing gain-scheduled controllers are given in terms of the feasible set of these LMIs. The improvement offered by our approach over existing methods for stability analysis and gain-scheduled controller synthesis for parameter-dependent linear systems are analyzed in theory. Numerical examples demonstrate that our approach can offer significant improvement in practice.

Memory state feedback control synthesis for linear systems with time delay via a finite number of linear matrix inequalities

In this paper, we consider a synthesis problem of delay-dependent memory state feedback control which stabilizes linear time-delay systems. First we derive conditions for stability analysis and controller synthesis in the form of infinite-dimensional (parameter-dependent) linear matrix inequalities (LMIs), while infinite dimensionality of the LMIs may lead to less conservative results, but makes the conditions difficult to use. Second we show a technique to reduce the infinite-dimensional LMIs to a finite number of LMIs. A numerical example is given to demonstrate out approach.

Augmented gradient flows for on-line robust pole assignment via state and output feedback

This paper is concerned with robust pole assignment in synthesis of linear control systems via state and output feedbacks. First, both the pole assignment and robustness requirements are appropriately formulated as two optimization problems. Then, gradient flow models are developed for the on-line computation of feedback gain matrices that result in robust pole assignment by solving these two optimization problems. A technique is introduced to facilitate the real-time matrix inverse involved for realizing the gradient flow models. The resulting augmented gradient flows have desired convergence properties. Simulation results are included to show the effectiveness of the proposed approach.

Memory State Feedback Control Synthesis for Linear Systems With Time Delay via a Finite Number of Linear Matrix Inequalities

In this paper, we consider a synthesis problem of delay-dependent memory state feedback control which stabilizes linear time-delay systems. We derive conditions for stability analysis and controller synthesis in the form of infinite-dimerisional (parameter dependent) Linear Matrix Inequalities LMIs and reduce the infinitedimensional LMIs to a finite number of LMIs. A numerical example is given to demonstrate its efficacy.

Computational algorithms for linear control systems: a brief survey

This paper presents a brief survey of computational algorithms for the analysis and synthesis of linear control systems described in the state space. An attempt is made to select the most efficient methods for analysis of the stability, controllability and observability, the reduction into canonical forms, the pole assignment synthesis and the synthesis of optimal systems with quadratic cost. Some aspects of the development of mathematical software for solving these problems are also discussed.

A Numerical Algorithm for the Most Economical Structural Synthesis of Linear Control Systems

A Numerical Algorithm for the Most Economical Structural Synthesis of Linear Control Systems

Synthesis of Linear Control Systems via Matrix Fraction Representation-1

Synthesis of Linear Control Systems via Matrix Fraction Representation-1

Synthesis of Linear Control Systems via Matrix Fraction Representation-3

Synthesis of Linear Control Systems via Matrix Fraction Representation-3

Synthesis of Linear Control Systems via Matrix Fraction Representation-2

Synthesis of Linear Control Systems via Matrix Fraction Representation-2

Control of Linear Systems Via the Serial Canonical Form

In the present paper an efficient computational algorithm for synthesis of linear control systems with any attainable by state feedback linearly equivalent form of the closed loop system is proposed. The algorithm is based on the serial canonical form of linear systems which is alternative to the canonical form of Brunovsky-Luenberger. The transformation of linear systems into the serial canonical form is based on orthogonal and diagonal matrix transformations only. A computer program for the realization of the method is developed and tested for systems of order up to 40.

On the synthesis of linear control systems with specified characteristics

The Popov invariants are used to define two matrices M AB and K AB , which describe the properties of a system x ̇ = Ax + Bu in a basis-independent form. Also a control system with state feedback K and nonsingular input transformation M, x ̇ = (A − BK)x + BMw = Fx + Gw , can be uniquely specified by the corresponding matrices M FG and K FG , which may be chosen arbitrarily with the only condition, that ( F, G) must have the same unordered set of Kronecker invariants as ( A, B). It is shown, how K and M are computed from M AB , K AB , M FG and K FG . Implications for pole assignment, minimal polynomial assignment and observer design are shown. Also a solution to the minimal realization problem is given, which results in M AB , K AB and the output relation y = Cx.

On The Synthesis of Linear Control Systems with Specified Characteristics

The invariants introduced by Popov are used to define two matrices MAB and KAB, which describe the properties of a system ẋ = Ax + Bu in a basis-independent form. Also a control system ẋ = (A−BK)x + BMw = Fx + Gw, which is obtained by state feedback Κ and nonsingular input transformation M, can be uniquely specified by the corresponding matrices MFG and Kfg which may be chosen arbitrarily with the only condition, that (F,G) must have the same unordered set of Kronecker invariants as (A,B). It is shown, how Κ and M are computed from MAB, KAB, MFG and KFG. Implications for pole assignment, minimal polynomial assignment and observer design are shown. Also a solution to the minimal realization problem is given, which immediately gives MAB, KAB and the output relation y = Cx.

Robust solutions to linear multivariable control problems

In this paper a special case of the Regulator Problem with Internal Stability (RPIS) [1] is studied from the standpoint of obtaining solutions that are insensitive to perturbations in both the system parameters and the controller gains. The ability to include robustness into the problem yields new insight into the role played by dynamic compensation in linear control system synthesis.