L1 analysis and design of sampled-data systems

The focus of this work is L₁-optimal control of sampled-data systems. A converging approximation procedure is derived to compute the L_∞-induced norm of closed-loop finite-dimensional linear time-invariant (LTI) sampled-data systems. An approximation method is developed to synthesize L₁-optimal sampled-data regulators. Finally, an example is provided that illustrates the L₁ analysis and design techniques presented     

L2-gain analysis of nonlinear systems andnonlinear state-feedback H∞ control

Previously obtained results on L2-gain analysis of smooth nonlinear systems are unified and extended using an approach based on Hamilton-Jacobi equations and inequalities, and their relation to invariant manifolds of an associated Hamiltonian vector field. On the basis of these results a nonlinear analog is obtained of the simplest part of a state-space approach to linear H_∞ control, namely the state feedback H_∞ optimal control problem. Furthermore, the relation with H_∞ control of the linearized system is dealt with     

Rational approximation of L1 optimalcontrollers for SISO systems

The L₁ optimal control problem with rational controllers for continuous-time systems is considered in which it is shown that the optimal L₁ performance index with rational controllers is equal to that of irrational controllers. A sequence of rational controllers that approximates the optimal index is constructed. Convergence properties of such a sequence are studied. That the corresponding sequence of objective transfer functions is shown to converge in weak-* topology in BV(R₊) in the time domain and uniformly in a wider sense in the frequency domain     

H2-optimal sampled-data control

The H₂-optimal control of continuous-time linear time-invariant systems by sampled-data controllers is discussed. Two different solutions, state space and operator theoretic, are given. In both cases, the H₂ sampled-data problem is shown to be equivalent to a certain discrete-time H₂ problem. Other topics discussed include input-output stability of sampled-data systems, performance recovery in digital implementation of analog controllers, and sampled-data control of systems with the possibility of multiple-time delays     

The robust H2 control problem: a worst-casedesign

The worst-case effect of a disturbance system on the H ₂ norm of the system is analyzed. An explicit expression is given for the worst-case H₂ norm when the disturbance system is allowed to vary over all nonlinear, time-varying and possibly noncausal systems with bounded L₂-induced operator norm. An upper bound for this measure, which is equal to the worst-case H₂ norm if the exogeneous input is scalar, is defined. Some further analysis of this upper bound is done, and a method to design controllers which minimize this upper bound over all robustly stabilizing controllers is given. The latter is done by relating this upper bound to a parameterized version of the auxiliary cost function studied in the literature     

On L1 optimal control

It is shown that the problem of minimizing the effect of the worst disturbance on the output, where the signals have finite one norms, is equivalent to the same problem with bounded disturbances. The following situations provide the motivation to study such a problem. (1) Disturbances are generated by a fixed bounded-input, bounded-output (BIBO) stable, linear time-invariant (LTI) system with unknown initial conditions and zero input. (The output of this fixed system is the disturbance input to the original control system). (2) Disturbances are generated by an unknown BIBO stable LTI system whose fixed input consists of the unit impulse (this is the dual of (1)). In both of these situations, the disturbances will have a finite one norm     

Two properties of l1-optimal controllers

The solution of the l₁ sensitivity minimization problem is shown to have two properties which contrast markedly with properties of the solutions to the better-known H∞ sensitivity minimization problem or the LQG (linear quadratic Gaussian) problem is given of a one-parameter family of first-order plants where the order of the l₁-optimal compensator can be arbitrarily large, and thus it is impossible to bound the order of an l₁-optimal compensator in terms of the order of the plant. A plant is considered which has a continuous one-parameter family of l₁-optimal compensators, and thus l₁-optimal compensators need not be unique. The author's two examples are considered to answer two questions left open by M.A. Daleh and J.B. Pearson (ibid., vol.32, p.314-23, 1987)     

Mixed H2/H∞ control:a convex optimization approach

The problem of finding an internally stabilizing controller that minimizes a mixed H₂/H_∞ performance measure subject to an inequality constraint on the H_∞ norm of another closed-loop transfer function is considered. This problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint. Both the state-feedback and output-feedback problems are considered. It is shown that in the state-feedback case one can come arbitrarily close to the optimal (even over full information controllers) mixed H₂/H_∞ performance measure using constant gain state feedback. Moreover, the state-feedback problem can be converted into a convex optimization problem over a bounded subset of (n×n and n ×q, where n and q are, respectively, the state and input dimensions) real matrices. Using the central H_∞ estimator, it is shown that the output feedback problem can be reduced to a state-feedback problem. In this case, the dimension of the resulting controller does not exceed the dimension of the generalized plant     

Optimal L∞ bounds for disturbancerobustness

Robust disturbance rejection bounds obtained recently by Zhu et al. are tight with respect to the disturbance set that has a specified L₂ norm bound. It is shown that the bounds are also tight with respect to a weighted L₂ disturbance set that has a specified bound on its outer product by choosing a special weight matrix in the L₂ norm. The matrix can computed by numerical iteration     

Minimization of the L∞-induced norm forsampled-data systems

It is shown that given any degree of accuracy, there exists a standard discrete-time l¹ problem that can be determined a priori whose solution yields a controller that is almost optimal in terms of the hybrid L^∞-induced norm. This is accomplished by first converting the hybrid system into an equivalent infinite-dimensional discrete-time system using the lifting technique in continuous time, and then approximating the infinite-dimensional parts of the system which model the intersample dynamics. A thorough analysis of the approximation procedure is presented, and it is shown that it is convergent at the rate of 1/n . Explicit bounds that are independent of the controller are obtained to characterize the approximation. It is also shown that the geometry of the induced norm for the sampled-data problem is different from that of the standard l¹ norm, and hence there might not exist a linear isometry that maps the sampled-data problem exactly to a standard discrete-time problem     

CACSD using the state-space L∞ theory-adesign example

The L^∞ optimal control theory is used to achieve singular value loop-shaping design objectives in a multivariable aircraft pitch-axis control design example involving an unstable nonminimum phase plant. Comparison to the frequency-weighted LQG (Linear quadratic Gaussian) design shows that the L^∞ theory easily produced a superior design having both higher bandwidth and greater stability margin, with no need for designer interaction for iterative adjustment of weights, than LQG. The design example provides a vivid illustration of how the all-pass property of the L^∞ theory enables precise shaping of closed-loop singular-value Bode plots to conform pointwise to prespecified weighting functions     

Necessary and sufficient conditions for mixedH2 and H∞ optimal control

It is shown that D.S. Bernstein and W.M. Hadad's (ibid., vol.34, no.3, p.293, 1989) necessary condition for full-order mixed H ₂ and H_∞ optimal control is also sufficient, and that J.C. Doyle et al.'s (Proc. Amer. Control Conf., p.2065, 1989) sufficient condition for full-order mixed H₂ and H_∞ optimal control is also necessary. They are duals of one another     

Disturbance-generated bifurcation in a prototype adaptive systemwith e1-modification law

The e₁-modification law has been proposed by K.S. Narendra and A.M. Annaswamy (ibid., vol.32, no.2, pp.134-5, 1987) for robust model reference adaptive control in the presence of bounded disturbances. Using this law, it is shown by the present authors that, in the case of regulation, a prototype adaptive control system with a single unknown pole and an unknown constant disturbance at its input possesses multiple equilibria and undergoes various bifurcations. Simulations and local analysis give a complete characterization of the possible dynamics exhibited by the disturbance adaptive system. As the constant disturbance is varied, saddle-node bifurcation, (subcritical) Hopf bifurcation, and saddle-connection bifurcation occur in that order. Consequently, in this case, the e₁-modification law is qualitatively equivalent to the σ-modification law     

A treatment of jw-axis model-matching transformation zerosin the optimal H2 and H∞control designs

A model-matching transformation (MMT) zero is defined as a rank-deficiency condition which prevents an H² or H^∞ optimal control problem from being transformed into an equivalent model-matching problem. By imposing saturation constraints and accounting for additive instrument noise in the sensor and actuator signals, all MMT zeros can be eliminated     

On a weaker notion of controllability of a language K withrespect to a language L

A necessary and sufficient condition for a prefix-closed language K⊆Σ* to be controllable with respect to another prefix-closed language L⊆Σ* is that K⊆ L. A weaker notion of controllability where it is not required that K⊆L is considered here. If L is the prefix-closed language generated by a plant automaton G, then essentially there exists a supervisor Θ that is complete with respect to G such that L(Θ|G)=K ∩ L if and only if K is weakly controllable with respect to L. For an arbitrary modeling formalism it is shown that the inclusion problem is reducible to the problem of deciding the weaker notion of controllability. Therefore, removing the requirement that K ⊆ L from the original definition of controllability does not help the situation from a decidability viewpoint. This observation is then used to identify modeling formalisms that are not viable for supervisory control of the untimed behaviors of discrete-event dynamic systems     

Stability of x(t)=Ax(t)+Bx(t-τ)

A stability criterion for linear time-delay systems described by a differential difference equation of the form dx(t)=Ax(t)+Bx(t -τ) is proposed. The result obtained includes information on the size of the delay and therefore can be a delay-dependent stability condition. Its relation to existing delay-independent stability criteria is also discussed     

Adaptive RLS algorithms under stochasticexcitation-L2 convergence analysis

A very general class of RLS (recursive least squares) algorithms having a forgetting factor is considered. The basic assumptions are that the data generation mechanism is free of disturbances and that the observation vector is a stochastic process satisfying a φ-mixing condition. A stochastic characterization of persistent excitation is first given. Then, it is proved that the algorithm is exponentially convergent in the mean-square sense     

G2 blend surfaces and filling ofN-sided holes

An approach to obtaining curvature-continuous blends of any parametrically given surface is presented. The approach offers several options to control the shape of the blend through geometric continuity. It lets the designer define linkage curves arbitrarily as parametric curves in the respective parameter domains. Moreover, the proposed transfinite interpolant can be adopted to fill an n-sided polygonal hole by a single patch     

H∞-optimal control with state-feedback

A H_∞-optimal control problem in which the measured outputs are the states of the plant is considered. The main result shows that the infimum of the norm of the closed-loop transfer function using linear static state-feedback equals the infimum of the norm of the closed-loop transfer function over all stabilizing dynamic (even, nonlinear time-varying) state-feedback controllers     

Controller reduction by H∞-balancedtruncation

H_∞-balanced truncation may be used to obtain reduced-order plants or controllers. The plant (possibly unstable) is compensated using a particular robustly stabilizing controller. The two Riccati equations involved are then used to define a set of closed-loop input-output invariants called the H_∞-characteristic values. That part of the plant or controller corresponding to small H_∞-characteristic values is discarded to give a reduced-order plant or controller. By exploiting an intimate connection with coprime factorization, a simple a priori test is derived for the ability of such a reduced-order controller to stabilize the full-order plant. The performance of the resulting closed-loop may also be bounded a priori, i.e. in terms of the prespecified level of robustness and the discarded H_∞-characteristic values