Stabilization of linear multivariable systems by output feedback

Absfract-A method is developed for improving the stability of linear multivariable systems using output feedback. The technique, which utilizes a gradient approach, has been mechanized in a digital computer program. Illustrative results are given for a seven-state two-feedback model of the Saturn V booster.     

The effect of feedback on linear multivariable systems

This paper describes effective algebraic procedures for performing certain operations on matrix transfer functions, the most important being the calculation of the effect of feedback. Such operations are required, for example, in designing, sequentially, controllers for linear multivariable systems. Previous papers have described algorithms for performing these operations numerically at specific frequencies to obtain, for example, the closed-loop matrix frequency response, in numerical form. However, if algebraic solutions, which are matrix transfer functions whose elements are rational functions, are required, naive use of standard formulae must be avoided, since they result in rational functions of needlessly high degree. This paper shows how this needless increase in the degree of the rational functions may be avoided, thus yielding effective algebraic procedures for the operations considered. Although the results are of interest in their own right, a brief resume of a specific procedure, the sequential return difference method, for designing linear multivariable control systems which utilises these results, is given.     

Algebraic theory of linear multivariable feedback systems

This paper presents an algebraic theory for analysis and design of linear multivariable feedback systems. The theory is developed in an algebraic setting sufficiently general to include, as special cases, continuous and discrete time systems, both lumped and distributed. Designs are implemented by construction of a controller with two vector inputs and one vector output. Use of controllers of this type is shown to generate convenient stability results, and convenient global parametrizations of all I/O maps and all disturbance-to-output maps achievable, for a given plant, by a stabilizing compensator. These parametrizations are then used to show that any such I/O map and any such disturbance-to-output map may be simultaneously realized by choice of an appropriate controller. In the special case of lumped systems, it is shown that the design theory. can be reduced to manipulations involving polynomial matrices only. The resulting design procedure is thus shown to be more efficient computationally. Finally, the problem of asymptotically tracking a class of input signals is considered in the general algebraic setting. It is shown that the classical results on asymptotic tracking can be generalized to this setting. Additionally, sufficient conditions for robustness of asymptotic tracking, and robustness of stability are developed.     

Eigenvalue assignment in linear multivariable systems by output feedback

The assignment of arbitrary cloacd-loop eigenvalue spectra by output feedback is investigated in the special ease of systems for which the set of state-feedback matrices P which yield a desired closed-loop eigenvalue spectrum is finite and can be readily generated. In particular, a simple necessary and sufficient condition for the existence of an output-feedback matrix K corresponding to a prescribed state-feedback matrix P is established which also leads directly to the matrix K in cases when the condition is satisfied. The condition derived in this paper requires neither the computation of generalized inverse matrices (Munro and Vardulakis 1973), nor the transformation of state vectors (Patel 1974), nor the determination of Luenberger canonical forms (Vardulakis 1976), and indicates very clearly that an output-feedback matrix K corresponding to a prescribed state-feedback matrix P exists only in very special circumstances.     

Dyadic feedback laws for linear multivariable systems

The extent to which it is possible to assign the eigenvalues of the system Where K is constrained to have the dyadic structure. K = qp^T, is investigated. It is shown that a necessary and sufficient condition for the existence of a dyadic feedback law to assign arbitrary eigenvalues, in so for as this can be accomplished by any state variable feedback law, is that the system matrix of the controllable part of the system is non-derogatory. The proof furnishes a constructive procedure for computing a vector q in this case. The treatment is based on the Anderson-Luenberger approach, and a now result is presented regarding the information which can be extracted from their canonical form.     

Decoupling of linear multivariable systems by unity output feedback compensation

The existence and the synthesis of a stabilizing compensator for the diagonal decoupling of a linear multivariable system is examined. A new sufficient condition is established which is generically satisfied by any proper plant P and guarantees the existence of a stabilizing compensator that will give rise to a closed-loop transfer function matrix which is non-singular and diagonal.     

On the robustness of multivariable linear feedback systems

One of the useful indicators of the robustness of a multivariable linear feedback system is the largest singular value of the nominal closed-loop transfer matrix. It is shown that while comparing different, not necessarily diagonal, closed-loop transfer matrices which have the same diagonal elements, the diagonal closed-loop transfer matrix has the greatest robustness. For plants with not "too" large parameter uncertainty, this result also guarantees the maximization of disturbance rejection, and the minimization of the control signal "power" at the plant's input.     

Identification of linear, multivariable systems operating under linear feedback control

The possibility of estimating process parameters using input-output data collected when the system operates in closed loop is discussed in this paper. Concepts that are useful for a systematic treatment of the problem are introduced. The results refer to the case where the regulator is a linear feedback law or alternates between several such laws. It is shown that a straightforwardly applied identification scheme has the same identifiability properties as the more complex method in which the parameters of the closed-loop system are estimated first. It is also shown that it is always possible to achieve the same identifiability properties as for open-loop systems by shifting between different linear regulators. The required number of regulators depends only on the number of inputs and outputs. The results obtained are illustrated by a numerical example.     

Identifiability conditions for linear multivariable systems operating under feedback

The possibility of estimating the parameters of a dynamic system when it is operating in closed loop is examined. Earlier considered ways of designing regulators to achieve desired identifiability properties are unified and generalized. The result of the analysis of this short paper gives a simple criterion, which contains earlier known conditions as simple special cases.     

Robust state-derivative feedback LMI-based designs for multivariable linear systems

In some practical problems, for instance in the control systems for the suppression of vibration in mechanical systems, the state-derivative signals are easier to obtain than the state signals. New necessary and sufficient linear matrix inequalities (LMI) conditions for the design of state-derivative feedback for multi-input (MI) linear systems are proposed. For multi-input/multi-output (MIMO) linear time-invariant or time-varying plants, with or without uncertainties in their parameters, the proposed methods can include in the LMI-based control designs the specifications of the decay rate, bounds on the output peak, and bounds on the state-derivative feedback matrix K. These design procedures allow new specifications and also, they consider a broader class of plants than the related results available in the literature. The LMIs, when feasible, can be efficiently solved using convex programming techniques. Practical applications illustrate the efficiency of the proposed methods.     

Complete decoupling of linear multivariable systems by means of linear static and differential state feedback

The concept of complete decoupling is introduced in this paper. We show in particular that for any linear controllable system with non-singular transfer matrix, there always exists a static and differential state feedback such that the closed-loop system is completely decoupled.     

Algorithm for closed-loop eigenstructure assignment by state feedback in multivariable linear systems

In this paper an algorithm is presented which greatly facilitates the complete exploitation of state feedback in the assignment of the entire closed-loop eigenstructure of controllable multi-input systems. This algorithm is a generalization of the algorithm of MacLane and Birkhoff (1968) for the computation of a basis for the null space of a matrix and is ideally suited to digital computer implementation. The algorithm readily yields the vectors which are required (Porter and D'Azzo 1978) for the simultaneous assignment of Jordan canonical forms, eigenvectors, and generalized eigenvectors to the plant matrices of closed-loop controllable multivariable linear systems. The effectiveness of the algorithm is illustrated by assigning the entire closed-loop eigenstructure of a third-order two-input discrete-time system in such a way that the resulting closed-loop system exhibits time-optimal behaviour.     

Computation of optimal output feedback gains for linear multivariable systems

For linear systems with quadratic performance indices, it is shown that the optimal output feedback gains can be computed using gradient techniques. Unlike previous algorithms, this approach avoids the solution of nonlinear matrix equations while appearing to ensure convergence. Computational results for a fourth-order system are presented.     

Nonlinear control of linear multivariable systems via state-dependent feedback gains

It is often desirable to have controllers which respond fast to large errors and which respond slowly to the small errors that are often due to sensor noise. Some nonlinear controllers having these properties are developed here by modifying optimal linear state feedback controllers so that they have state-dependent gains. A general approach is developed for the nonlinear control of multivariable systems such that the transient responses exhibit the desired properties, and the closed-loop systems are asymptotically stable.     

Minimum sensitivity design of linear multivariable feedback control systems by matrix spectral factorization

A scalar measure of system sensitivity to plant parameter variations is employed in the design of linear lumped stationary multivariable feedback control systems. The plant parameters are treated as random variables, and design formulas are derived which lead to systems with the smallest expected value for the chosen scalar sensitivity measure. The design formulas give physically realizable feedback and tandem compensation network transfer function matrices provided the overall system transfer function matrix is properly specified. The solution of the minimum sensitivity design problem is obtained by first solving the multivariable semi-free-configuration Wiener problem.     

Time-scale structure assignment in linear multivariable systems using high-gain feedback

A systematic design method for assigning a required time-scale structure to a given multivariable system is developed. The time-scale structure assignment is carried out using high-gain feedback of either the state or the output. The method developed decomposes a given multivariable system into several single-input single-output subsystems, each of which is designed separately. A standard method of design for single-input single-output systems is developed, and this forms a building block for the multivariable system design. The state feedback design depends only on the stage sequence of the given system, while the output feedback design depends only on the set of integers that specifies the order of infinite zeros. In this way, our design is made robust with respect to certain parameter uncertainties. High-gain parameter(s) can be adjusted on line, and this provides a tool for the fine tuning of the desired performance characteristics of a control system.     

Internal stabilization and decoupling in linear multivariable systems by unity output feedback compensation

We consider the problem of internally stabilizing and simultaneously diagonally decoupling a linear multivariable system by unity output feedback compensation. A sufficient condition is derived for the existence of a cascade proper compensatorC(s)such that when employed in a unity feedback loop involving the proper transfer function matrix Poof a free of unstable hidden modes systemSigma(P_{o}), will not only internally stabilize the feedback closed-loop systemSigma(P_{o}, C)but will also give rise to a closed-loop transfer function matrixH_{yr}^{diag}, which is nonsingular, diagonal, and has desired poles. Based on this analysis, an algorithmic procedure for the computation of such a compensator is presented.     

The stability of linear multivariable systems

A general criterion is presented for establishing the stability of a multivariable system from the stability of the simplified system consisting of its diagonal elements. This new criterion includes all known criteria of stability based on the diagonal system elements such as Rosenbrock's (1974), Nwokah's (1980) anil Koussiouris's (1980). For the class of matrices satisfying this criterion an equivalent result in terms of eigenloci can also be stated     

On the robustness of multivariable linear feedback systems in state-space representation

A singular values-based approach to specify the robustness of a multivariable linear feedback system in state-space representation is investigated. The robustness measure which is considered is the largest spectral norm of an additive uncertainty in the closed-loop system matrix, for which stability is guaranteed. It is shown that under the constraint of prescribed pole placement, a lower bound for the robustness measure is maximized, when the Frobenius norm of the closed-loop system matrix is minimized.     

The design of linear multivariable systems

This paper describes a computer-aided procedure whereby a succession of single-loop designs, using Nyquist loci, yields a multivariable design which is stable and attenuates disturbances. The procedure permits more freedom than previously available for the selection of the compensating matrix. It is shown how a simple modification of the procedure enables security against component failure to be obtained.