Design of multivariable feedback systems with stable plant

This paper considers, in a general algebraic framework, the design of a unity-feedback multivariable system with a stable plant. The method is based on a simple parameterization of the four closed-loop transfer functions in terms ofP, the plant transfer function, andQ=H_{y_{1}u_{1}}. In particular, the I/O transfer functionQ=H_{y_{2},u_{1}}=PQ. Using the framework of rational transfer functions, we show that the closed-loop system will be exponentially stable if and only ifQis exponentially stable. Furthermore, if bothPandQare strictly proper then the controller is also strictly proper. The basic result is Design Theorem 2. An algorithm is given for obtaining strictly proper controllers such that the resulting I/O map is decoupled, all its poles can be chosen by the designer, and the same holds for zeros except, of course, for the C+-zeros prescribed by the C+-zeros of the plant. A discussion is included to temper these results by the constraints imposed by noise and plant saturation.     

Cancellations in multivariable continuous-time and discrete-time feedback systems treated by greatest common divisor extraction

This correspondence considers a multivariable system with proper rational matrix transfer functions G0and Gfin the forward and feedback branches, respectively. It develops a strictly algebraic procedure to obtain polynomials whose zeros are the poles of the matrix transfer functions from input to output (Hy), and from input to error (He). G0and Gfare given in the polynomial matrix factored formN_{0}D_{0}^{-1}andD_{f}^{-1}N_{f}. The role of the assumption det [I + G_{f}(infty)G_{0}(infty)] neq 0and the relation between the zeros of det [I + G_{f}G_{0}] and the poles of Hyand Heare indicated. The implications for stability analysis of continuous-time as well as discrete-time systems are stressed.     

On the design of single-loop single-input-output feedback control systems in the complex-frequency domain

This paper is concerned with single-loop, single-input-output, feedback control systems consisting of a plant, controller, and feedback sensor each modeled by a rational transfer function. The plant can be unstable, nonminimum phase, and even improper. The objective is the determination of the controller transfer function for which the system is asymptotically stable and design objectives are met. This has always been the goal in feedback system design. What is novel about the approach presented here is that the family of all stabilizing controller transfer functions is parameterized in terms of a rational functionK (s). The design method allows for complete freedom in the selection of all closed-loop system poles (except those associated with hidden modes). This is accomplished through the choice of the denominator polynomial forK (s). The remaining design freedom lies in the selection of the numerator ofK (s). This freedom is utilized to shape the system sensitivity function and to achieve specified steady-state error coefficients. A numerical example in which the plant is unstable and nonminimum phase is presented to illustrate the above points. The final design is compared with alternative designs and the tradeoffs are clearly displayed.     

Control of decentralized systems with distributed controller complexity

An approach is presented which allows design of stabilizing decentralized controllers for linear systems with two scalar channels, such that each local controller is endowed with some dynamics while the sum of the orders is kept smaller than the system order. It is shown that for almost allnth-order systems with two scalar channels the local controllers can be chosen such that their orders δ1and δ2satisfydelta_{1} +delta_{2}leq n - 2, max{delta_{1}, delta_{2}} leq max {(n - 1)/2, n -1 - (mu_{0}/2)}, whereμ0is the number of stable zeros of the cross Coupling transfer functions in the system. The approach is to design one of the local controllers such that the McMillan degree of the resulting one-channel system is reduced. Then the other local controller only has to deal with a model of reduced dimension and can thus be chosen of lower order.     

Zeros of f(z) = (az-b)npm (cz-d)n

The zeros off(z) = (az - b)^{n} pm (cz - d)^{n}are found to lie on a circle of radius|(ad - cb)/(|a|^{2} - |c|^{2})|with its center atz = (a^{ast}b - c^{ast}d)/(|a|^{2} - |c|^{2}), wherea, b, c, anddare complex numbers andnis assumed real. When|a| = |c|the locus of the zeros is a straight line perpendicular to the line joining the pointsb/aandb/cand intersecting it atz = 0.5(b/a + d/c). The zeros are found analytically and constructed geometrically.     

A new approach to the solution of the l1 controlproblem: the scaled-Q method

We explore an approach for solving multiple input-multiple output (MIMO) l₁ optimal control problems. This approach, which we refer to as the scaled-Q approach, is introduced to alleviate many of the difficulties facing the numerical solution of optimal l₁ control problems. In particular, the computations of multivariable zeros and their directions are no longer required. The scaled-Q method also avoids the pole-zero cancellation difficulties that existing methods based on zero-interpolation face when attempting to recover the optimal controller from an optimal closed-loop map. Because the scaled-Q approach is based on solving a regularized auxiliary problem for which the solution is always guaranteed to exist, it can be used no matter where the system zeros are (including the stability boundary). Upper and lower bounds that converge to the optimal cost are provided, and all solutions are based on finite dimensional linear programming for which efficient software exists     

Parametric models of linear multivariable systems for adaptive control

This paper presents a parametrization method for direct adaptive control of linear multivariable systems with strictly proper discrete-time or continuous-time transfer functions. The necessary a priori information is shown to be a diagonal matrix with the noninvertible zeros of the Smith form and appropriate polynomial degrees.     

Existence of dominant solutions in linear output feedback

The standard LQ-regulator is known to be a dominant controller in the sense that the optimal cost is minimal for all initial statesx_{0} in R^{n}. In general, static or dynamic output feedback controllers do not have this dominance property. In this note, it is shown that in the general multivariable case and for the original cost functional, a dynamic output feedback controller using an observer is dominant if, and only if, the observer is perfectly initialized.     

Adaptive control with recursive identification for stochastic linear systems

This paper presents a stochastic adaptive control algorithm which is shown to possess the following properties when applied to a possibly unstable, inverse stable, linear stochastic system with unknown parameters, whenever that system satisfies a certain positive real condition on its (moving average) noise dynamics. 1) The adaptive control part of the algorithm stabilizes and asymptotically optimizes the behavior of the system in the sense that the (limit of the) sample mean-square variation of the-output around a given demand level equals that of a minimum variance control strategy implemented with known parameters. This optimal behavior is subject to an offset μ²where μ²is the variance of a dither signal added to the control action in order to produce a "continually disturbed control." Formu^{2} > 0, it is shown that the input-output process satisfies a persistent excitation property, and hence, subject to a simple identifiability condition, the next property holds. 2) The observed input and output of the controlled system may be taken as inputs to an approximate maximum likelihood algorithm (AML) which generates strongly consistent estimates of the system's parameters. Results are presented for the scalar and multivariable cases.     

On the minimum order of a robust servocompensator

In recent papers [1]-[4], Davison et al. have given the characterization of a minimal-order robust error-driven servocompensator which achieves asymptotic tracking and disturbance rejection. In this note, we establish this minimality property by frequency domain methods. We show that any right coprime factorization of the controller, sayN_{r}(S)D_{r}(S)^{-1}, must have all the elements ofD_{r}(s)divisible byPhi(s), the minimal polynomial of the tracking and disturbance signal generator. Hence, its order must be at leastn_{0}.d(phi)(n0=number of outputs,d(Phi)= degree ofPhi).     

Multivariable feedback, sensitivity, and decentralized control

In this paper the problem of sensitivity, reduction by feedback is studied and related to a problem of decentralized control. A plant will be represented by anN times Nmatrix of frequency responses, which may be unstable or irrational. The object will be to find conditions onP(s)under which a diagonal feedbackF(s)can make the sensitivityparallel{I + P(s)F(s)}^{-1}parallelarbitrarily small over some specified frequency interval [-jomega_{0}, jomega_{0}] without violating a global sensitivity, boundparallel{I+ P(s)F(s)}^{-1}parallel leq M, (Mgeqsome const. >1) forRe(s) geq 0. It will be shown that such a diagonal feedback of the "high gain" type can be constructed wheneverP^{-1}(s)is analytic inRe(s)geq 0, P(s)satisfies an attenuation condition nears = infty, andP(s)approaches diagonal dominance at high frequencies. It will also be shown that these conditions on the plant can be interpreted as conditions for the existence of a decentralized wide-band control scheme.     

Robust adaptive fault tolerant control for a process with actuator faults

This paper studies design and implementation of an enhanced multivariable adaptive control scheme for an uncertain nonlinear process exposed to actuator faults. For adaptive fault compensation, a model reference adaptive control (MRAC) strategy is utilized as main controller. A new adaptation algorithm making possible to improve transient performance of adaptive control is integrated to the controller. With the help of further modifications, some restrictive conditions on multivariable adaptive design are relaxed so that the system requires less plant information. The resulting controller has a simpler structure than the other matrix factorization based controllers. At the final stage of design, a robust adaptive control scheme is obtained with consideration of practical implementation problems such as sensor noises, external disturbances and unmodeled​ system dynamics. It is proved that the controller guarantees closed-loop signal boundedness and asymptotic output tracking. Real-time experiment results acquired from quadruple tank benchmark system are presented in order to exhibit the effectiveness of the proposed scheme.     

On the structure of maximal(A, B)-Invariant subspaces: A polynomial matrix approach

Given a controllable and observable triple (A, B, C) describing a linear time invariant multivariable system Σ, which gives rise to a full rank transfer function matrixT_{o}(s), the structure of the maximal (A, B)- invariant subspace contained inker Cis investigated using a polynomial matrix approach. Thus, certain connections between the geometric and the polynomial matrix approaches to linear system theory are established.     

Adaptive Neural Network Tracking Control With Disturbance Attenuation for Multiple-Input Nonlinear Systems

A switching adaptive neural network controller for multiple-input nonlinear, affine in the control dynamical systems with unknown nonlinearities is designed, capable of arbitrarily attenuating $L_{2}$ or $L_{infty} $ external disturbances. In the absence of disturbances, a uniform ultimate boundedness property of the tracking error with respect to an arbitrarily small set around the origin is guaranteed, as well as the uniform boundedness of all the signals in the closed loop. The proposed switching adaptive controller effectively avoids possible division by zero, while guaranteeing the continuity of switching. In this way, problems connected to existence of solutions and chattering phenomena are alleviated. Simulations illustrate the approach.      

Improvement of stability of zeros in discrete-time multivariable systems using fractional-order hold

This paper is concerned with the stability of zeros of the discrete-time multivariable systems composed of a fractional-order hold (FROH), a continuous -time plant and a sampler in cascade. The properties of the limiting zeros are studied and a condition for obtaining stable zeros for sufficiently small sampling periods is derived. An approximate fractional-order hold (AFROH) is proposed as an implementation of FROH. It is shown that the AFROH as well as the FROH can locate the zeros of discrete-time multivariable systems inside the stable region and improve stability properties when the zero-order hold (ZOH) cannot. Experimental results of adaptive control with AFROH demonstrate a significant improvement of control performance compared with the use of a ZOH.     

应用单层神经网络设计多变量自适应模糊控制器

提出一种应用单层神经网络设计多变量自适应模糊控制器的方法。应用单层神经网络可以学习多变量模糊控制规则中的未知参数，还可由它来实现多变量模糊推理过程。该方法能解决多变量模糊控制中普遍存在的规则获取困难和难于实现实时自适应等问题。仿真试验表明，所设计的多变量模糊控制器不仅实时性好，而且可得到满意的控制效果。     

On the identification of state-derivative-coupled systems

This short paper treats one aspect of the identification of state-derivative-coupled systems, such asMdot{x}(t) = Ax(t) + Bu(t) + w(t)whereM neq I, andMis invertible. This equation can also be written asdot{x}(t) = F_{1}x(t) + F_{2}u(t) + omega(t). We assume that reduced form parameters (F_{1}, F_{2}) are identifiable and develop a sequence of tests for establishing the identifiability of structural parameters (M, A, B) from (F_{1} F_{2}). The tests are constructive, in that they not only can be used to ascertain the identifiability of (M, A, B); but, if (M, A, B) are not identifiable, can also indicate corrective actions to be taken so that (M, A, B) are identifiable.     

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.     

Feedback properties of multivariable systems: The role and use of the return difference matrix

For linear time-invariant multivariable feedback systems, the feedback properties of plant disturbance attenuation, sensor noise response, stability margins, and sensitivity to plant and sensor variation are quantitatively related to the Bode magnitude versus frequency plots of the singular values of the return difference matrixI + Land of the associated inverse-return difference matrixI + L^{-1}. Implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed. The penalty function in the stochastic linear quadratic Gaussian (LQG) optimal control problem is found to be a weighted-sum of the singular values, with the weights determined by the quadratic cost and noise intensity matrices. This enables systematic "tuning" of LQG cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties. The theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft.     

Design of cascade controllers for zero assignment in multivariable systems†

The paper describes a method for the design of a constant cascade controller to assign some zeros and numerator coefficients of the (open or closed-loop) transfer function matrix of a linear multivariable system. The cascade controller does not affect the poles of the system which can be positioned prior to the zero assignment. The requirements on the transfer function matrix for arbitrary assignment of some zeros and numerator coefficients are discussed. The cascade controller is calculated from sets of simple linear equations and thus the method is computationally very efficient. An interactive computer programme is described for the computer-aided design of cascade controllers. Finally, two simple numerical examples of zero assignment in open-loop and closed-loop systems are worked out in detail to illustrate the method.