On a generalization of the Youla–Kučera parametrization. Part II: the lattice approach to MIMO systems

Within the lattice approach to analysis and synthesis problems recently developed in Quadrat (Signal Syst, to appear), we obtain a general parametrization of all stabilizing controllers for internally stabilizable multi input multi output (MIMO) plants which do not necessarily admit doubly coprime factorizations. This parametrization is a linear fractional transformation of free parameters and the set of arbitrary parameters is characterized. This parametrization generalizes for MIMO plants the parametrization obtained in Quadrat (Syst Control Lett 50:135–148, 2003) for single input single output plants. It is named general Q-parametrization of all stabilizing controllers as we show that some ideas developed in this paper can be traced back to the pioneering work of Zames and Francis (IEEE Trans Automat control 28:585-601, 1983) on H _∞-control. Finally, if the plant admits a doubly coprime factorization, we then prove that the general Q-parametrization becomes the well-known Youla-Kučera parametrization of all stabilizing controllers (Desoer et al. IEEE Trans Automat control 25:399–412, 1980; Vidyasagar, Control system synthesis: a factorization approach MIT Press, Cambridge 1985).     

Further results on global stabilization of uncertain nonlinear systems by output feedback

For a family of uncertain nonlinear systems dominated by a triangular system that satisfies linear growth condition with an output dependent growth rate, we prove that global robust stabilization can be achieved by smooth output feedback. This conclusion has incorporated and generalized the recent output feedback stabilization results, for instance, the work (IEEE Trans. Automat. Control 2002; 47 :2068–2073) where the same conclusion was already shown to be true for planar systems, and the work (Proceedings of the 42nd IEEE, CDC, 2003; 1544–1549) where the growth rate is required to be a polynomial function of the system output. There are two key ingredients in the present contribution. One of them is the introduction of a rescaling transformation with a dynamic factor that is tuned on‐line through a Riccati‐like differential equation, which turns out to be extremely effective in dealing with the system uncertainty. The other one is the development of a recursive observer design algorithm making it possible to assign the robust observer gains in a step‐by‐step fashion. Both a smooth state feedback controller and a robust observer are explicitly constructed for the rescaled system using only the knowledge of the bounding system. Copyright © 2005 John Wiley & Sons, Ltd.     

A note on static output‐feedback for affine nonlinear systems

In this paper we present sufficient conditions for the solvability of the static‐output feedback stabilization problem for affine nonlinear systems using a factorization approach. We extend the results of (IEEE Trans. Autom. Control, Vol. 47, No. 12, pp. 2038–2041 (2002)) to include disturbance‐attenuation and singular control. The sufficient conditions given are also less stringent than the ones given in (IEEE Trans. Autom. Control Vol. 47, No. 12, pp. 2038–2041 (2002)). The usefulness of the results are also illustrated with some examples. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Observers for a class of Lipschitz systems with extension to performance analysis

In this paper, observer design for a class of Lipschitz nonlinear dynamical systems is investigated. One of the main contributions lies in the use of the differential mean value theorem (DMVT) which allows transforming the nonlinear error dynamics into a linear parameter varying (LPV) system. This has the advantage of introducing a general Lipschitz-like condition on the Jacobian matrix for differentiable systems. To ensure asymptotic convergence, in both continuous and discrete time systems, such sufficient conditions expressed in terms of linear matrix inequalities (LMIs) are established. An extension to H_∞ filtering design is obtained also for systems with nonlinear outputs. A comparison with respect to the observer method of Gauthier et al. [A simple observer for nonlinear systems. Applications to bioreactors, IEEE Trans. Automat. Control 37(6) (1992) 875–880] is presented to show that the proposed approach avoids high gain for a class of triangular globally Lipschitz systems. In the last section, academic examples are given to show the performances and some limits of the proposed approach. The last example is introduced with the goal to illustrate good performances on robustness to measurement errors by avoiding high gain.     

A Kharitonov‐like theorem for robust stability independent of delay of interval quasipolynomials

In this paper, a Kharitonov‐like theorem is proved for testing robust stability independent of delay of interval quasipolynomials, p(s)+∑eq_k(s), where p and q_k's are interval polynomials with uncertain coefficients. It is shown that the robust stability test of the quasipolynomial basically reduces to the stability test of a set of Kharitonov‐like vertex quasipolynomials, where stability is interpreted as stability independent of delay. As discovered in (IEEE Trans. Autom. Control 2008; 53 :1219–1234), the well‐known vertex‐type robust stability result reported in (IMA J. Math. Contr. Info. 1988; 5 :117–123) (See also (IEEE Trans. Circ. Syst. 1990; 37 (7):969–972; Proc. 34th IEEE Conf. Decision Contr., New Orleans, LA, December 1995; 392–394) does contain a flaw. An alternative approach is proposed in (IEEE Trans. Autom. Control 2008; 53 :1219–1234), and both frequency sweeping and vertex type robust stability tests are developed for quasipolynomials with polytopic coefficient uncertainties. Under a specific assumption, it is shown in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) that robust stability independent of delay of an interval quasipolynomial can be reduced to stability independent of delay of a set of Kharitonov‐like vertex quasipolynomials. In this paper, we show that the assumption made in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) is redundant, and the Kharitonov‐like result reported in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) is true without any additional assumption, and can be applied to all quasipolynomials. The key idea used in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) was the equivalence of Hurwitz stability and ?_‐o‐stability for interval polynomials with constant term never equal to zero. This simple observation implies that the well‐known Kharitonov theorem for Hurwitz stability can be applied for ?_‐o‐stability, provided that the constant term of the interval polynomial never vanishes. However, this line of approach is based on a specific assumption, which we call the CNF‐assumption. In this paper, we follow a different approach: First, robust ?_‐o‐stability problem is studied in a more general framework, including the cases where degree drop is allowed, and the constant term as well as other higher‐orders terms can vanish. Then, generalized Kharitonov‐like theorems are proved for ?_‐o‐stability, and inspired by the techniques used in (IEEE Trans. Autom. Control 2008; 53 :1219–1234), it is shown that robust stability independent of delay of an interval quasipolynomial can be reduced to stability independent of delay of a set of Kharitonov‐like vertex quasipolynomials, even if the assumption adopted in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) is not satisfied. Copyright © 2009 John Wiley & Sons, Ltd.     

On the controllability of the parameterized state matrix in “a continuation approach to eigenvalue assignment” by Harris,DeCarlo and Richter

This paper will investigate the controllability properties for systems parameterized as in Harns et al. (1983, Automatica, 19, 551–555). It will be shown that for systems of low dimensions this parameterization must be done carefully to guarantee that the system is controllable over the parameterization. The controllability property is key to the algorithms developed in Lefebvre et al. (1985, Int. J. Control, 41, 1273–1292), Richter and DeCarlo (1984, IEEE Trans. Aut. Control, AC-29), Harris et al. (1983, Automatica, 19, 551–555), Richter et al. (1981, Proc. 1981 Joint Aut. Control Conf.), Harris and DeCarlo (1982, Purdue School of EE, TR-EE 82-11) and Sebok and DeCarlo (1984, Proc. Allerton Conf. Communication, Control and Computing).     

Non-resonance conditions for uniform observability in the problem of nonlinear output regulation

This paper deals with the design of an internal model-based semiglobal output feedback regulator for possibly non-minimum phase nonlinear systems. Taking advantage of the design tool proposed in Isidori (IEEE Trans. Automat. Control 45 (10) 2000), we show how the problem of output regulation can be reformulated into an output feedback stabilization problem of a suitably defined extended auxiliary system. The output feedback stabilization of the extended auxiliary system is addressed in the second part of the paper, where an observer-based stabilizer is proposed. The existence of the latter is characterized in terms of necessary and sufficient conditions which can be interpreted as nonlinear non-resonance conditions between the modes of the exosystem and the zero dynamics of the controlled plant.     

Stabilization of a synchronous generator with a controllable series capacitor via immersion and invariance

The immersion and invariance (I&I) technique (it IEEE Trans. Automat. Control 2003; 48 (4):590–606) is a recently proposed control methodology for stabilizing nonlinear systems. Here we apply this philosophy to stabilize a single machine infinite bus (SMIB) system using a controllable series capacitor (CSC). The synchronous generator is modeled using the second‐order swing equation and a first‐order model is used for the CSC. The control objective here is to immerse a desired second‐order dynamic model into the higher order system manifold and design an asymptotically stabilizing control law. Copyright © 2011 John Wiley & Sons, Ltd.     

A dual‐observer design for global output feedback stabilization of nonlinear systems with low‐order and high‐order nonlinearities

This paper employs a dual‐observer design to solve the problem of global output feedback stabilization for a class of nonlinear systems whose nonlinearities are bounded by both low‐order and high‐order terms. We show that the dual‐observer comprised of two individual homogeneous observers, can be implemented together to estimate low‐order and high‐order states in parallel. The proposed dual observer, together with a state feedback controller, which has both low‐order and high‐order terms, will lead to a new result combining and generalizing two recent results (Li J, Qian C. Proceedings of the 2005 IEEE Conference on Decision and Control, 2005; 2652–2657) and (Qian C. Proceedings of the 2005 American Control Conference, June 2005; 4708–4715). Copyright © 2008 John Wiley & Sons, Ltd.     

Nonlinear small-gain theorems for discrete-time feedback systems and applications

We derive in this work a local nonlinear small-gain theorem in the framework of input-to-state stability for discrete time systems. Our primary objective is to show that, as in the continuous-time context, these discrete-time nonlinear small-gain theorems are very effective in stability analysis and synthesis for various classes of discrete-time control systems. Two converse Lyapunov theorems for discrete exponential stability are developed to assist these applications. New results in stability and stabilization presented in this paper are significant extensions of previous work by other authors (IEEE Trans. Automat. Control 38 (1993) 1398; 39 (1994) 2340; 33 (1988) 1082).     

Design of nonlinear observers with approximately linear error dynamics using multivariable Legendre polynomials

This paper presents a numerical approach to the design of nonlinear observers by approximate error linearization. By using a Galerkin approach on the basis of multivariable Legendre polynomials an approximate solution to the singular PDE of the observer design technique proposed by Kazantzis and Krener (see (Syst. Control Lett. 1998; 34 :241–247; SIAM J. Control Optim. 2002; 41 :932–953)) is determined. It is shown that the L₂‐norm of the remaining nonlinearity in the resulting error dynamics can be made small on a specified multivariable interval in the state space. Furthermore, a linear matrix equation is derived for determining the corresponding change of co‐ordinates and output injection such that the proposed design procedure can easily be implemented in a numerical software package. A simple example demonstrates the properties of the new numerical observer design. Copyright © 2006 John Wiley & Sons, Ltd.     

Hybrid feedback stabilization of systems with quantized signals

This paper is concerned with global asymptotic stabilization of continuous-time systems subject to quantization. A hybrid control strategy originating in earlier work (Brockett and Liberzon, IEEE Trans. Automat. Control 45 (2000) 1279) relies on the possibility of making discrete on-line adjustments of quantizer parameters. We explore this method here for general nonlinear systems with general types of quantizers affecting the state of the system, the measured output, or the control input. The analysis involves merging tools from Lyapunov stability, hybrid systems, and input-to-state stability.     

On the design of output regulators for nonlinear systems

This paper considers the output regulation problem for nonlinear systems. By adding the assumption that the decoupling matrix of the system under consideration has maximum rank, the general results presented in Isidori and Byrnes (IEEE Trans. Automat. Control 35(2) (1990) 131–140) have been improved in three ways. Firstly, the precise form of the regulators is presented for multi-input multi-output affine nonlinear systems. Secondly, the number of unknown functions in the controller parameterization has been reduced. Finally, by introducing a hyperbolic realization and constructing auxiliary systems, a sufficient and generically necessary condition for the existence of the unknown functions is obtained. This condition is easily verifiable using only algebraic computation. These results provide a complete solution to the output regulation problem, for a class of nonlinear systems. It is also shown that for this class of systems, the output regulation problem is always generically solvable.     

State feedback stabilization of stochastic nonlinear systems with SiISS inverse dynamics

This paper further considers a more general class of stochastic nonlinear systems with stochastic integral input‐to‐state stability (SiISS) inverse dynamics and drift and diffusion terms depending upon the other states besides stochastic inverse dynamics and the first state. By skillfully choosing the designed functions and the update laws of parameters, and using the important mathematical tools established in IEEE Trans. Automat. Contr. 2010; 55 (2):304–320, a unifying framework of state feedback controller is proposed to guarantee that all the signals of the closed‐loop system are bounded almost surely and the states can be regulated to zero almost surely. A simulation example demonstrates the effectiveness of the control scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive nonlinear control with partial overparametrization

The main purpose of this paper is to show that overparametrization in adaptive nonlinear control is not always a bad thing and can be intentionally introduced to achieve similar performance with less control effort. To this end, an adaptive controller with partial overparametrization is first designed to solve the regulation problem of parametric-strict-feedback form nonlinear systems by combining the design procedure in Krstic et al. (System Control Lett. 19 (1992) 177) with that in Kanellakopoulos et al. (IEEE Trans. Automat. Control 36 (1991) 1241). Then, to reduce the control effort needed by the above controller further, a modified adaptive controller is designed by introducing a reference signal and reformulating the regulation problem as a tracking problem. The simulation results show that, compared with the adaptive controller without overparametrization, the adaptive controller with partial overparametrization can achieve similar performance with less control effort, and the modified one can do the same thing with even less control effort.     

A C‐OWA operator‐based approach to decision making with interval fuzzy preference relation

Yager [IEEE Trans Syst Man Cybern B 2004;34:1952–1963] introduced a continuous interval argument OWA (C‐OWA) operator, which extends the ordered weighted averaging (OWA) operator, introduced by Yager [IEEE Trans Syst Man Cybern B 1988;18:183–190], to the case in which the given argument is a continuous valued interval rather than a finite set of values. In this article, we utilize the C‐OWA operator to derive the priority vector of an interval fuzzy preference relation and then develop a practical approach to solving the decision‐making problem with interval fuzzy preference relation. Finally, a numerical example is provided to demonstrate the practicability and efficiency of the developed approach. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 1289–1298, 2006.     

Periodic motion planning for virtually constrained Euler–Lagrange systems

The paper suggests an explicit form of a general integral of motion for some classes of dynamical systems including n-degrees of freedom Euler–Lagrange systems subject to (n-1) virtual holonomic constraints. The knowledge of this integral allows to extend the classical results due to Lyapunov for detecting a presence of periodic solutions for a family of second order systems, and allows to solve the periodic motion planning task for underactuated Euler–Lagrange systems, when there is only one not directly actuated generalized coordinate. As an illustrative example, we have shown how to create a periodic oscillation of the pendulum for a cart–pendulum system and how then to make them orbitally exponentially stable following the machinery developed in [A. Shiriaev, J. Perram, C. Canudas-de-Wit, Constructive tool for an orbital stabilization of underactuated nonlinear systems: virtual constraint approach, IEEE Trans. Automat. Control 50 (8) (2005) 1164–1176]. The extension here also considers time-varying virtual constraints.     

Nonlinear H∞ observer design for one‐sided Lipschitz discrete‐time singular systems with time‐varying delay

This paper investigates the H_∞ observer design problem for a class of nonlinear discrete‐time singular systems with time‐varying delays and disturbance inputs. The nonlinear systems can be rectangular and the nonlinearities satisfy the one‐sided Lipschitz condition and quadratically inner‐bounded condition, which are more general than the traditional Lipschitz condition. By appropriately dealing with these two conditions and applying several important inequalities, a linear matrix inequality–based approach for the nonlinear observer design is proposed. The resulting nonlinear H_∞ observer guarantees asymptotic stability of the estimation error dynamics with a prescribed performance γ. The synthesis condition of H_∞ observer design for nonlinear discrete‐time singular systems without time delays is also presented. The design is first addressed for one‐sided Lipschitz discrete‐time singular systems. Finally, two numerical examples are given to show the effectiveness of the present approach.     

A reduced-order observer for non-linear discrete-time systems

In this note, a simple and useful reduced-order observer for a large class of non-linear discrete-time systems is presented. Neither detectability of the linear part nor resolution of the Sylvester constraint is required. From the stability analysis we show that the proposed state observer has strong tracking properties under very general conditions and generalizes results in (M. Boutayeb et al., IEEE. Trans. Automat. Control 42(4) (1997) 581–586). High performances, from the design and implementation point of view, will be shown through three numerical examples of physical processes largely used in industrial applications.     

Robust output feedback stabilization via risk-sensitive control

We consider a problem of robust linear quadratic Gaussian (LQG) control for discrete-time stochastic uncertain systems with partial state measurements. For a finite-horizon case, the problem was recently introduced by Petersen et al. (IEEE Trans. Automat. Control 45 (2000) 398). In this paper, an infinite horizon extension of the results of Petersen et al. (IEEE Trans. Automat. Control 45 (2000) 398) is discussed. We show that for a broad class of uncertain systems under consideration, a controller constructed in terms of the solution to a specially parameterized risk-sensitive stochastic control problem absolutely stabilizes the stochastic uncertain system.