The optimal projection equations for reduced-order state estimation: The singular measurement noise case

The optimal projection equations for reduced-order state estimation are generalized to allow for singular (i.e., colored) measurement noise. The noisy and noise-free measurements serve as inputs to dynamic and static estimators, respectively. The optimal solution is characterized by necessary conditions which involve a pair of oblique projections corresponding to reduced estimator order and singular measurement noise intensity.     

U-D factorisation of the strengthened discrete-time optimal projection equations

Algorithms for optimal reduced-order dynamic output feedback control of linear discrete-time systems with white stochastic parameters are U-D factored in this paper. U-D factorisation enhances computational accuracy, stability and possibly efficiency. Since U-D factorisation of algorithms for optimal full-order output feedback controller design was recently published by us, this paper focusses on the U-D factorisation of the optimal oblique projection matrix that becomes part of the solution as a result of order-reduction. The equations producing the solution are known as the optimal projection equations which for discrete-time systems have been strengthened in the past. The U-D factored strengthened discrete-time optimal projection equations are presented in this paper by means of a transformation that has to be applied recursively until convergence. The U-D factored and conventional algorithms are compared through a series of examples.     

Robust, reduced-order, nonstrictly proper state estimation via theoptimal projection equations with guaranteed cost bounds

A state estimation design problem involving parametric plant uncertainties is considered. An estimation error bound suggested by multiplicative white-noise modeling is utilized for guaranteeing robust estimation over a specified range of parameter uncertainties. Necessary conditions that generalize the optimal projection equations for reduced-order state estimation are used to characterize the estimator that minimizes the error bound. The design equations thus effectively serve as sufficient conditions for synthesizing robust estimators. Additional features include the presence of a static estimation gain in conjunction with the dynamic (Kalman) estimator to obtain a nonstrictly proper estimator     

Robust, reduced-order, nonstrictly proper state estimation via the optimal projection equations with Petersen—Hollot bounds

A state-estimation design problem involving parametric plant uncertainties is considered. An error bound suggested by recent work of Petersen and Hollot is utilized for guaranteeing robust estimation. Necessary conditions which generalize the optimal projection equations for reduced-order state estimation are used to characterize the estimator which minimizes the error bound. The design equations thus effectively serve as sufficient conditions for synthesizing robust estimators. An additional feature is the presence of a static estimation gain in conjunction with the dynamic (Kalman) estimator, i. e., a nonstrictly proper estimator.     

线性不确定系统的给定阶最优控制器设计

为避免间接法设计降阶控制器的模型近似引起的性能下降,本文在静态输出反馈控制器设计的基础上,直接设计了线性不确定系统的给定阶混合H2/H∞动态反馈控制器.利用系统内外分解方法,得到了最优降阶状态观测器.通过求解降维状态观测器的静态输出反馈,可得到降阶控制的最优反馈增益阵.给定阶控制器由两个Ric cati方程和一个Lyapunov方程参数化表示.最后,通过一个例子,说明了本文提出的给定阶控制器设计方法.     

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.     

Frequency domain design of reduced order observer based H ∞ controllers - a polynomial approach

In this paper a frequency domain solution of the singular H X control problem, where s of the m measurements y ( t ) of the plant are not affected by disturbances, is considered. By applying the polynomial approach to the parameterization of state feedback and of reduced order observers the singular H X control problem is solved for a-n n th order plant using a frequency domain representation of a reduced order observer based H X controller of order n- s . This output feedback controller is directly calculated in the frequency domain by first solving the H X state feedback problem and then solving the singular H X filtering problem for a modified system model, which involves the J -spectral factorization of two polynomial matrix equations. The results presented are obtained by considering the time domain solution first providing good insight in the relations between the time and frequency domain approach to the solution of the singular H X control problem using reduced order observer based controllers.     

Stationary LQG control of singular systems

The stationary linear-quadratic-Gaussian control problem is formulated and solved for single-input single-output singular systems. The control system is required to be internally proper and stable in order to avoid both impulsive and unstable exponential behavior. The set of all controllers resulting in such a control system is specified in parametric form. All controllers that yield finite cost are identified, once again in parametric form, within this set. Necessary and sufficient conditions are then established for an optimal controller to exist. All optimal controllers are shown to possess the same transfer function. The problem is analyzed in the complex domain. The transfer functions are expressed as quotients of proper, strict-Hurwitz rational functions. By means of this maneuver, the powerful tools of algebra are made available. The synthesis of the optimal controller is reduced to the solution of two linear Diophantine equations whose coefficients are obtained by spectral factorization.     

Design of dynamic suboptimal low-order controllers on basis of the H∞-criterion

A design problem of dynamic reduced-dimension controllers according to the boundedness condition of the H_∞-norm of a transfer matrix of a closed system is studied. The problem of controller deflation is related to the solution to degenerate filtering problems (no noise of measuring) and to degenerate control problems (no control in a manipulated output). It was demonstrated that these problems can be solved on basis of the modified 2-Riccati approach, in which the “filtrating” (for a singular filtrating problem) or the “control” (for a singular control problem) Lurie-Riccati equations have a low order. An example of an optimal reduced-dimension controller design is given that illustrates the obtained results.     

A periodic fixed-structure approach to multirate control

We develop an approach to designing reduced-order multirate controllers. A discrete-time model that accounts for the multirate timing sequence of measurements is presented and is shown to have periodically time-varying dynamics. Using discrete-time stability theory, the optimal projection approach to fixed-order (i.e. full- and reduced-order) dynamic compensation is generalized to obtain reduced-order periodic controllers that account for the multirate architecture. It is shown that the optimal reduced-order controller is characterized by means of a periodically time-varying system of equations consisting of coupled Riccati and Lyapunov equations. In addition, the multirate static output-feedback control problem is considered. For both problems, the design equations are presented in a concise, unified manner to facilitate their accessibility for developing numerical algorithms for practical applications     

The optimal projection equations with Petersen-Hollot bounds:robust stability and performance via fixed-order dynamic compensationfor systems with structured real-valued parameter uncertainty

A feedback control design problem involving structured real-valued plant parameter uncertainties is considered. A quadratic Lyapunov bound suggested by recent work of I.R. Petersen and C.V. Hollot (1986) is utilized in conjunction with the guaranteed cost approach of S.S.L. Chang and T.K.C. Peng (1972) to guarantee robust stability with robust performance bound. Necessary conditions that generalize the optimal projection equations for fixed-order dynamic compensation are used to characterize the controller that minimizes the performance bound. The design equations thus effectively serve as sufficient conditions for synthesizing dynamic output-feedback controllers that provide robust stability and performance     

Output deadbeat controllers with stability for discrete-time multivariable linear systems

This short paper Treats the problem of designing output deadbeat controllers having the property that the control input to the system converges to zero as time goes to infinity, for discrete-time multivariable linear systems. Two configurations of controllers are considered: one is of state feedback; the other is a dynamic controller using an observer. The existence of such controllers is examined, and the methods are presented for designing such controllers when they exist. The controller using a state feedback obtained in this paper is optimal in the sense that the controller settles the output in zero for any initial state in the minimum number of steps. On the other hand, the dynamic controller is not optimal in that sense, but it minimizest, wheretis defined as an integer such that the controller drives the output to zero in no more thantsteps for any set of initial conditions of the system and the observer.     

The optimal reduced-order estimator for systems with singularmeasurement noise

The optimal reduced-order estimator is completely characterized by necessary conditions, resulting from the optimal projection equations. The solution consists of one Riccati equation and two Lyapunov equations coupled by two projections. Explicit expressions for all of the estimator parameters are given. The relation between the reduced-order singular estimator and the full-order optimal singular estimator (which is of reduced order itself) is investigated. It is shown that under certain conditions the optimal estimator is recovered from the reduced-order estimator     

Robust stabilization of singularly perturbed systems

We discuss the problem of designing stabilizing controllers for singularly perturbed systems on the basis of simplified models. In [1], it was shown that a constant gain output feedback controller designed on the basis of the simplified model need not stabilize the ‘true’ system containing both fast and slow modes. This phenomenon was then expanded to include the case where the simplified system is strictly proper in [2]. The objectives of this note are threefold: (i) to show that, given any proper system and any stabilizing controller for it that is proper but not strictly proper, there exists a singular perturbation of the system that is destabilized by that controller, (ii) to show that any strictly proper controller for a singularly perturbed system designed on the basis of a reduced order model will stabilize the true system for sufficiently small values of the fast dynamics parameter, and (iii) to provide a characterization, in the same spirit as [3,4], of the set of all strictly proper controllers that stabilize a given proper plant. By combining these results, it is possible to generate the class of all robustly stabilizing controllers for a given singularly perturbed system.     

Robustness of linear feedback control systems to unmodelled high-frequency dynamics

A basic issue in the control of linear time-invariant plants is the effect of neglected high-frequency dynamics on the performance, in particular on the closed-loop stability, of the control system. In this paper, the robustness of various output feedback control designs, based on a reduced-order model with neglected high-frequency dynamics, is investigated using singular perturbation techniques. A general robust design rule is to avoid static output feedback for systems with unmodelled high-frequency dynamics. From a frequency-domain standpoint, the robust design rule is to avoid closing high-frequency plant loops by using strictly proper controllers or controllers with a low-pass filtering property.     

Data-Driven Global Robust Optimal Output Regulation of Uncertain Partially Linear Systems

In this paper, a data-driven control approach is developed by reinforcement learning (RL) to solve the global robust optimal output regulation problem (GROORP) of partially linear systems with both static uncertainties and nonlinear dynamic uncertainties. By developing a proper feedforward controller, the GROORP is converted into a global robust optimal stabilization problem. A robust optimal feedback controller is designed which is able to stabilize the system in the presence of dynamic uncertainties. The closed-loop system is ensured to be input-to-output stable regarding the static uncertainty as the external input. This robust optimal controller is numerically approximated via RL. Nonlinear small-gain theory is applied to show the input-to-output stability for the closed-loop system and thus solves the original GROORP. Simulation results validates the efficacy of the proposed methodology.      

The optimal projection equations for reduced-order, discrete-time state estimation for linear systems with multiplicative white noise

The optimal projection equations obtained in [2,3] for reduced-order, discrete-time state estimation are generalized to include the effects of state- and measurement-dependent noise to provide a model of parameter uncertainty. In contrast to the single matrix Riccati equation arising in the full-order (Kalman filter) case, the optimal steady-state reduced-order discrete-time estimator is characterized by three matrix equations (one modified Riccati equation and two modified Lyapunov equations) coupled by both an oblique projection and stochastic effects.     

Robustness of dynamic output feedback control for singular perturbation systems

The problem of the robustness of dynamic output feedback control for singular perturbation systems is investigated. The solution to this problem is reduced to the simultaneous design of static output feedback controllers for the fast subsystem and the so-called auxiliary system of the slow subsystem. Some conditions are proposed to ensure the robustness of the actual closed-loop system. The exact upper bound of the parasitic parameter for the controlled system is also determined. Finally, an actual model which failed in dynamic output feedback control in [4] is reexamined successfully here.     

Optimal control of Takagi–Sugeno fuzzy-model-based systems representing dynamic ship positioning systems

Orthogonal function approach (OFA) and the hybrid Taguchi-genetic algorithm (HTGA) are used to solve quadratic finite-horizon optimal controller design problems in both a fuzzy parallel distributed compensation (PDC) controller and a non-PDC controller (linear state feedback controller) for Takagi–Sugeno (TS) fuzzy-model-based control systems for dynamic ship positioning systems (TS-DSPS). Based on the OFA, an algorithm requiring only algebraic computation is used to solve dynamic equations for TS-fuzzy-model-based feedback and is then integrated with HTGA to design quadratic finite-horizon optimal controllers for TS-DSPS under the criterion of minimizing a quadratic finite-horizon integral performance index, which is also converted to algebraic form by the OFA. Integration of OFA and HTGA in the proposed approach enables use of simple algebraic computation and is well adapted to the computer implementation. Therefore, it facilitates design tasks of quadratic finite-horizon optimal controllers for the TS-DSPS. The applicability of the proposed approach is demonstrated in the example of a moored tanker designed using quadratic finite-horizon optimal controllers.     

Non-fragile output feedback H∞ vehicle suspension control using genetic algorithm

This paper presents an approach to design static output feedback and non-fragile static output feedback H_∞ controllers for active vehicle suspensions by using linear matrix inequalities and genetic algorithms. A quarter-car model with active suspension system is considered in this paper. By suitably formulating the minimization problem of the sprung mass acceleration, suspension deflection and tyre deflection, a static output feedback H_∞ controller and a non-fragile static output feedback H_∞ controller are obtained. The controller gain is naturally constrained in the design process. The approach is validated by numerical simulation which shows that the designed static output feedback H_∞ controller can achieve good active suspension performance in spite of its simplicity, and the non-fragile static output feedback H_∞ controller has significantly improved the non-fragility characteristics over controller gain variations.