H∞ optimal and suboptimal controllers for infinitedimensional SISO plants

This paper presents a simple formula for H^∞ optimal and suboptimal controllers for unstable SISO distributed plants, with rational weighting functions. The controller is expressed in terms of i) inner and outer parts of the plant, ii) a finite dimensional spectral factor obtained from the weighting functions, and iii) a rational function satisfying certain interpolation conditions. Under certain genericity assumptions, this rational function is of dimension less than or equal to n₁+l-1 (n₁+l in the suboptimal case), where l is the number of unstable poles of the plant and n₁ is the order of the sensitivity weighting function. There are 2(n₁+l) (2(n₁+l+1) in the suboptimal case) linear equations, which determine this rational function. These linear equations can be written directly from the structure of the controller     

Array algorithms for H∞ estimation

We develop array algorithms for H^∞ filtering. These algorithms can be regarded as the Krein space generalizations of H ² array algorithms, which are currently the preferred method fur implementing H² filters. The array algorithms considered include two main families: square-root array algorithms, which are typically numerically more stable than conventional ones, and fast array algorithms which, when the system is time-invariant, typically offer an order of magnitude reduction in the computational effort. Both have the interesting feature that one does not need to explicitly check for the positivity conditions required for the existence of H^∞ filters, as these conditions are built into the algorithms themselves. However, since H^∞ square-root algorithms predominantly use J-unitary transformations, rather than the unitary transformations required in the H² case, further investigation is needed to determine the numerical behavior of such algorithms     

H∞ bounds for least-squares estimators

We obtain upper and lower bounds for the H^∞ norm of the Kalman filter and the recursive-least-squares (RLS) algorithm, with respect to prediction and filtered errors. These bounds can be used to study the robustness properties of such estimators. One main conclusion is that, unlike H^∞-optimal estimators which do not allow for any amplification of the disturbances, the least-squares estimators do allow for such amplification. This fact can be especially pronounced in the prediction error case, whereas in the filtered error case the energy amplification is at most four. Moreover, it is shown that the H^∞ norm for RLS is data dependent, whereas for least-mean-squares (LMS) algorithms and normalized LMS, the H^∞ norm is simply unity     

H∞ control for more general nonlinear systems

The authors consider the standard H^∞-control problem for more general nonlinear systems modeled by equations in which the penalty output and the measured output are, in general, functions of the state, the exogenous input, and the control input. In particular, we characterize a family of H^∞ controllers via output feedback as well as state feedback, solving the problem. The results obtained generalize some recent results in the literature     

Near-optimal H∞ control of linear singularlyperturbed systems

We consider the singularly perturbed H^∞ control problem under perfect state measurements, for both finite and infinite horizons. We suggest a construction of high-order approximations to a controller that guarantees a desired performance level on the basis of the exact decomposition of the full-order Riccati equations to the reduced-order slow and fast equations. This leads to effective asymptotic and numerical algorithms. We show that the high-order accuracy controller improves the performance     

H∞ control for linear time-varying systems:controller parameterization

In this correspondence, the H^∞ control problem is studied for finite-dimensional linear time-varying (FDLTV) systems. State-space formulas are derived for all FDLTV controllers solving the problem. These controllers are parameterized by the interconnection of the “central controller” with an exponentially stable, free system having H^∞ norm strictly less than γ. Our approach is drawn from some changes of variables, a time-varying version of a strict bounded real lemma, and Youla parameterization; thus, the proofs given are simple and clear     

H∞-optimal control for singularly perturbedsystems. II. Imperfect state measurements

For pt.I, see ibid., vol. 29, no. 2, p 401-423 (1993). In this paper we study the H^∞-optimal control of singularly perturbed linear systems under general imperfect measurements, for both finite- and infinite-horizon formulations. Using a differential game theoretic approach, we first show that as the singular perturbation parameter (say, ϵ>0) approaches zero, the optimal disturbance attenuation level for the full-order system under a quadratic performance index converges to a value that is bounded above by (and in some cases equal to) the maximum of the optimal disturbance attenuation levels for the slow and fast subsystems under appropriate “slow” and “fast” quadratic cost functions, with the bound being computable independently of E and knowing only the slow and fast dynamics of the system. We then construct a controller based on the slow subsystem only and obtain conditions under which it delivers a desired performance level even though the fast dynamics are completely neglected. The ultimate performance level achieved by this “slow” controller can be uniformly improved upon, however, by a composite controller that uses some feedback from the output of the fast subsystem. We construct one such controller, via a two-step sequential procedure, which uses static feedback from the fast output and dynamic feedback from an appropriate slow output, each one obtained by solving appropriate ϵ-independent lower dimensional H^∞-optimal control problems under some informational constraints. We provide a detailed analysis of the performance achieved by this lower-dimensional ϵ-independent composite controller when applied to the full-order system and illustrate the theory with some numerical results on some prototype systems     

Semi-decentralized adaptive fuzzy control for cooperativemultirobot systems with H∞ motion/internal forcetracking performance

We present a semi-decentralized adaptive fuzzy control scheme for cooperative multirobot systems to achieve H(infinity) performance in motion and internal force tracking. First, we reformulate the overall system dynamics into a fully actuated system with constraints. To cope with both parametric and nonparametric uncertainties, the controller for each robot consists of two parts: 1) model-based adaptive controller; and 2) adaptive fuzzy logic controller (FLC). The model-based adaptive controller handles the nominal dynamics which results in both zero motion and internal force errors for a pure parametric uncertain system. The FLC part handles the unstructured dynamics and external disturbances. An H(infinity) tracking problem defined by a novel performance criterion is given and solved in the sequel. Hence, a robust controller satisfying the disturbance attenuation is derived being simple and singularity-free. Asymptotic convergence is obtained when the fuzzy approximation error is bounded with finite energy. Maintaining the same results, the proposed controller is further simplified for easier implementation. Finally, the numerical simulation results for two cooperative planar robots transporting an object illustrate the expected performance.     

Parameter identification for uncertain linear systems with partialstate measurements under an H∞ criterion

This paper addresses the worst-case parameter identification problem for uncertain single-input/single-output (SISO) and multi-input/multi-output (MIMO) linear systems under partial state measurements and derives worst-case identifiers using the cost-to-come function method. In the SISO case, the worst-case identifier obtained subsumes the Kreisselmeier observer as part of its structure with parameters set at some optimal values. Its structure is different from the common least-squares (LS) identifier, however, in the sense that there is additional dynamics for the state estimate, coupled with the dynamics of the parameter estimate in a nontrivial way. In the MIMO case as well, the worst-case identifier has additional dynamics for the state estimate which do not appear in the conventional LS-based schemes. Also for both SISO and MIMO problems, approximate identifiers are obtained which are numerically much better conditioned when the disturbances in the measurement equations are “small”. The theoretical results are then illustrated on an extensive numerical example to demonstrate the effectiveness of the identification schemes developed     

Persistent identification and adaptation: stabilization of slowlyvarying systems in H∞

In this paper, the problem of persistent identification and adaptive stabilization of time-varying systems is studied within the framework of deterministic worst case identification and slow H^∞ adaptation. The plants under consideration are unstable and time-varying and cannot be stabilized by a fixed robust controller. Starting from an initial well-designed operating point, the controller must persistently adapt to the time-varying plant to maintain uniform stability over all future time. A key property which guarantees uniform stability is that the identification-adaptation iteration satisfies a certain invariance principle. We demonstrate that the adaptive design using periodic external inputs, least squares identification, and slow H^∞ adaptation possesses such an invariance property, leading to a successful adaptive stabilization methodology. Generic natures of our findings are discussed     

Comments on “Mixed L∞/H∞suboptimal controllers for SISO continuous time systems”

The authors state that they have no serious criticism of the original paper but make certain remarks. An example in the original illustrates a method to do tradeoffs between two design goals: H_∞ and L^∞. To make a fair comparison, they authors say that the optimal H_∞ norm should have been presented. The also point out misprints, and make remarks on the cancellation of the integrator pole and the tending to ∞ of the settling time     

An H∞/MinMax periodic control in a two-dimensionalstructural acoustic model with piezoceramic actuators

A feedback control computational methodology for reducing acoustic sound pressure levels in a two-dimensional cavity with a flexible boundary (a beam) is investigated. The control is implemented in this model through voltages to piezoceramic patches on the beam which are excited in a manner (out-of-phase) so as to produce pure bending moments. The incorporation of the output feedback control in this system leads to a problem with unbounded input and output terms. By writing the resulting system as an abstract Cauchy equation, the problem of reducing interior pressure levels can be posed in the context of an H_∞ /MinMax time-domain state-space formulation. A summary of extensive computational efforts comparing output feedback to full state feedback and investigating the effect of the number and location of sensors on performance of the control is presented     

A family of nonlinear H∞-output feedbackcontrollers

State-space formulas are derived for a family of controllers solving the nonlinear H^∞-output feedback control problem. The formulas given are expressed in terms of the solutions to two Hamilton-Jacobi inequalities in n independent variables. These controllers are obtained by interconnecting the “central controller” with an asymptotically stable, free system having L ²-gain ⩽γ. All proofs given are simple and clear and provide deeper insight in the synthesis of the corresponding linear H^∞ controllers     

Robust H∞-output feedback control of decoupledautomobile active suspension systems

H^∞-output feedback control is applied to the control of automobile active suspensions based on a dynamic model of the full vehicle. The output feedback control is desirable from the viewpoint of implementation in the sense that it reduces the number of measurements drastically compared with state feedback used currently in suspension control. In this paper, the authors show that a linearized model can be block-decoupled by a similarity transformation, which reduces the controller complexity significantly. The uncertainties of the vehicle model are properly taken into account in the derivation of the H^∞ controller. The controller is actually implemented in a commercial car, and the performance is evaluated both by simulations and experiments. The performance obtained proves to be quite satisfactory     

Rational L∞-suboptimal controllers for SISOcontinuous-time systems

We study the problem of minimizing the weighted amplitude of the time response due to a given fixed input signal for single-input/single-output (SISO) continuous-time systems and focus on obtaining rational suboptimal solutions. An EAS (Euler approximating system)-based method is proposed for designing a rational L_∞ -suboptimal controller for SISO systems. It is shown that this rational approximation is the best one among a set of rational approximations, in the sense of providing the tightest upper bound and that it can approximate the optimal cost arbitrarily close     

Alternating projection methods for mixed H2 andH∞ Nehari problems

In this paper, alternating convex projection methods along with fast Fourier transform techniques are used to solve some mixed H² and H^∞ Nehari problems     

H∞-optimal feedback controllers for linear multivariable systems

This paper treats the problem of designing, for a linear multivariable plant, a feedback controller which minimizes theH^{infty}-norm of a weighted sensitivity matrix. There exists a family of optimal improper feedbacks. This family is determined by application of a theory of Ball and Helton. A method for computing optimal feedbacks is described in detail and a numerical example is included. It is shown that an optimal improper feedback can be approximated by a proper one under certain conditions on the weighting matrices.     

Inverting and noninverting H∞ controllers

It is well known that two-block S/KS/T H_∞ problems in which the plant is weighted at the output tend to invert the plant in the controller. This paper shows that even four-block S/KS/T problems in which the plant is weighted at the input result in controllers which invert the plant. However, if a GS/T weighting scheme is used where the weight for the sensitivity includes the plant, the inversion is avoided. This GS/T scheme therefore is especially suited for ill-conditioned plants. An example confirms these results.     

Corrections to “On the structure of H∞control systems and related extensions”

The authors state that in the original paper (Kimura, Lu and Kawatani, ibid., vol. 36, p. 653-67, 1991) a lemma on (J,J')-losslessness is not correct. Its stated implications are also not fully correct. They give a counterexample to these claims and show how they can be fixed by strengthening the assumption     

一种分数阶内模TIλDemμ控制器设计方法

针对整数阶、 分数阶被控对象,提出了一种分数阶内模TI^λD^μ控制器的设计方法．首先对原被控模型进行降阶处理,得到二阶分数阶延时系统模型．然后将内模控制思想引入到降阶模型,设计内模控制器G_IMC（s）．最后将G_IMC（s）与所提的新型TI^λD^μ控制器进行参数对照,能够清晰地得出参数间的对应关系,从而得出分数阶TI^λD^μ的各参数．仿真结果表明,TI^λD^μ控制器能够获得良好的控制品质和鲁棒性．