Some results on the genericity of the pole placement problem

In this note, the pole placement problem for a linear MIMO systems with p outputs and m inputs is studied from the algebraic point of view. A formulation is proposed, that allows to analyze both theoretical and numerical aspects of the case min(m,p)=2 with more sharpness. Moreover, here it is shown that arbitrary pole placement by static output feedback of unitary rank is generically not possible even if m+p>n holds true.     

The Popov criterion and the inverse Nyquist method†

Kosonbrock's inverse Nyquist array (I.N.A.) theory for linear multivariable control Bystema with constant feedback elements is extended, to include systems lip to m nonlinear feedback elements, where the system has m inputs and m outputs. This extension is achieved by considering the Popov criterion for the most general case and through two further theorems. It shows that, as in the ease of Rosenbrock's I.N.A. theory, when certain auxiliary conditions are met with the help of suitable controllers, the design of multivariable controllers containing many non-linear feedbacks, can be based on the m frequency response loci corresponding to the diagonal entries of the open-loop inverse transfer function matrix. This leads to a simple design technique identical to the I.N.A. design method, suitable for use with a computer-aided design facility which permits a designer to use his intuitive understanding of transfer functions based on classical theory. The I.N.A. theory has been extended by Rosenbrock to cover systems having non-linear, time-varying feedback elements very recently.     

Fuzzy scheduling strategy for generalized switched server systems and its robustness over system heterogeneity

Generalized switched server system, a discretely controlled continuous‐time system, in which N tanks are used to represent N parallel entities, respectively, can be employed to address a class of load‐balancing problems. A tank‐pair model is a system that consists of two tanks and a single input single output controller, which regulates the inflows of the two tanks to acquire the two uniform levels under the specified inflow constraints. According to a quantized observation of the N tank levels, some discrete events are generated, and based on certain event feedback strategy, switching the location of the tank‐pair can control all the N tanks in a time‐sharing manner to acquire the N levels uniformity. Different from some existing scheduling strategies, this study proposes a fuzzy scheduling strategy (FSS) for such generalized switched server systems. Special measures are taken to reduce the N‐inputs two‐outputs fuzzy inference to a two‐inputs one‐output one, which greatly facilitates fuzzy scheduler design. Simulation results show that the proposed FSS strategy outperforms over the three existing scheduling strategies as a whole, and they also show that the proposed FSS strategy demonstrates high robustness over system heterogeneity. © 2009 Wiley Periodicals, Inc.     

LMI‐based second‐order sliding set design using reduced order of derivatives

Sliding mode control design for systems with relative degree r requires a number r ? 1 of time‐derivatives of the system output, which usually leads to deterioration of the whole scheme; if the highest‐order derivative is spared, a better precision is ensured. This paper proposes a control algorithm that guarantees reaching a second‐order sliding manifold using only r ? 2 derivatives of the system output. This objective is achieved at the price of yielding finite‐time convergence while preserving the essential feature of insensitivity to matched disturbances. The results take full advantage of convex representations and linear matrix inequalities, whose formulation easily allows dealing with unmatched disturbances by convex optimization techniques already implemented in commercially available software. Simulation examples are included to show the effectiveness of the proposed approach. Copyright © 2014 John Wiley & Sons, Ltd.     

A single dynamic observer-based module for design of simultaneous fault detection,isolation and tracking control scheme

The problem of simultaneous fault detection, isolation and tracking (SFDIT) control design for linear systems subject to both bounded energy and bounded peak disturbances is considered in this work. A dynamic observer is proposed and implemented by using the H_∞/H_?/L₁ formulation of the SFDIT problem. A single dynamic observer module is designed that generates the residuals as well as the control signals. The objective of the SFDIT module is to ensure that simultaneously the effects of disturbances and control signals on the residual signals are minimised (in order to accomplish the fault detection goal) subject to the constraint that the transfer matrix from the faults to the residuals is equal to a pre-assigned diagonal transfer matrix (in order to accomplish the fault isolation goal), while the effects of disturbances, reference inputs and faults on the specified control outputs are minimised (in order to accomplish the fault-tolerant and tracking control goals). A set of linear matrix inequality (LMI) feasibility conditions are derived to ensure solvability of the problem. In order to illustrate and demonstrate the effectiveness of our proposed design methodology, the developed and proposed schemes are applied to an autonomous unmanned underwater vehicle (AUV).     

Arbitrary Pole Placement by Constant Output Feedback for Linear Time Invariant Systems

In this paper, the pole placement problem by constant output feedback for linear time invariant systems is investigated. In particular, is proven that, for the class of linear time invariant systems with m inputs, p outputs, and McMillan degree n, the condition (m + p) > n is necessary for the solution of the arbitrary pole placement problem by constant output feedback.     

Fast-forward solver for inhomogeneous media using machine learning methods: artificial neural network,support vector machine and fuzzy logic

Encountering with a nonlinear second-order differential equation including ϵ _r and μ _r spatial distributions, while computing the fields inside inhomogeneous media, persuaded us to find their known distributions that give exact solutions. Similarities between random distributions of electric properties and known functions lead us to estimate them using three mathematical tools of artificial neural networks (ANNs), support vector machines (SVMs) and Fuzzy Logic (FL). Assigning known functions after fitting with minimum error to arbitrary inputs using results of machine learning networks leads to achieve an approximate solution for the field inside materials considering boundary conditions. A comparative study between the methods according to the complexity of the structures as well as the accuracy and the calculation time for testing of unforeseen inputs, including classification, prediction and regression is presented. We examined the extracted pairs of ϵ _r and μ _r with ANN, SVM networks and FL and got satisfactory outputs with detailed results. The application of the presented method in zero reflection subjects is exemplified.

     

Controllability and stability of repetitive batch processes

Batch control has traditionally addressed the problems related to the absence of a steady-state and the finite batch duration. Recently, batch control has added the dimension that arises naturally from the possibility of applying run-to-run control, i.e. it exploits the fact that most batch process are repeated over time. Hence, batch control calls for tools that are tailored to these new challenges and specificities of batch operations. These include a mathematical representation that explicitly shows the two independent time variables (the run time t and the run index k), two types of inputs (both constant and time-varying within a run) as well as two types of outputs (the run-time and run-end outputs). This paper introduces a definition of batch-output controllability that considers the two types of inputs and outputs. Also, a quantitative notion of stability that takes into account the finite-time nature of batch processes and typical phenomena such as the batch kick is presented. The tools required for evaluating these properties are readily adapted from the literature. As illustration, a semi-batch reactor example is considered in simulation. Various approaches to batch control are demonstrated and the associated controllability and stability issues discussed.     

Optimum Control of a Certain Class of Servomechanism†

Linear control systems governed by the vector matrix differential equation x = A x + B u have been considered. It has been shown how to find the optimum control u so that the system, starting from an initial position x(0), is steered to a state specifying the first p coordinates of the system in time t _o fixed in advance, the values attained by the (n—p) coordinates being immaterial, where n is the dimension of the system. The optimization considered here is with regard to the norm of u supposed to belong to L _m E _r space.     

Design of low-order delayed-measurement observers for discrete-time linear systems

This paper considers the development of a discrete-time low-order delayed-measurement observer for a discrete time-invariant linear system. The observer has several unique features. It uses discrete-time delayed measurements as part of its inputs and it is an rth-order observer for an nth-order linear system with q linearly dependent outputs, where r <n ? q. The purpose of this observer is to implement the control directly without estimating the state first. It is shown that under certain conditions the dimension of the observer is much lower than the standard observer of dimension n?q. The procedure employed here presents an approach which can be very important in large-scale control implementation.     

Regularity of distributional differential algebraic equations

Time-varying differential algebraic equations (DAEs) of the form E[(x)\dot]=Ax+f{E\dot{x}=Ax+f} are considered. The solutions x and the inhomogeneities f are assumed to be distributions (generalized functions). As a new approach, distributional entries in the time-varying coefficient matrices E and A are allowed as well. Since a multiplication for general distributions is not possible, the smaller space of piecewise-smooth distributions is introduced. This space consists of distributions which could be written as the sum of a piecewise-smooth function and locally finite Dirac impulses and derivatives of Dirac impulses. A restriction can be defined for the space of piecewise-smooth distributions, this restriction is used to study DAEs with inconsistent initial values; basically, it is assumed that some past trajectory for x is given and the DAE is activated at some initial time. If this initial trajectory problem has a unique solution for all initial trajectories and all inhomogeneities, then the DAE is called regular. This generalizes the regularity for classical DAEs (i.e. a DAE with constant coefficients). Sufficient and necessary conditions for the regularity of distributional DAEs are given.     

Collision-based computing implemented by soldier crab swarms

Collision-based computing schemes are implemented using propagating and interacting localisations. When two or more travelling localisations collide, they may reflect or merge into a new localisation. Presence/absence of a localisation in a specified site of a physical space represents True/False values of Boolean variables. Incoming trajectories symbolise inputs and outgoing trajectories are outputs. Logical gates are implemented by collisions between the localisations. We demonstrate how primitives of the collision-based computing can be implemented using swarms of soldier crabs in laboratory experiments. Soldier crabs, Mictyris guinotae, exhibit coherent collective behaviour on an intertidal flat at a daytime low-tide period. The soldier crabs' swarm often moves as a single entity with definite yet dynamically changing boundary. We utilise swarms of soldier crabs to implement a Boolean logic gate. The gate has two inputs and three outputs. Values of the inputs and outputs are represented by crab swarm size or a proportion of crabs in the input and output channels. We demonstrate that the gate performs logical conjunction and negation.     

Zero dynamics and funnel control of linear differential-algebraic systems

We study the class of linear differential-algebraic m-input m-output systems which have a transfer function with proper inverse. A sufficient condition for the transfer function to have proper inverse is that the system has ??strict and non-positive relative degree??. We present two main results: first, a so-called ??zero dynamics form?? is derived; this form is??within the class of system equivalence??a simple form of the DAE; it is a counterpart to the well-known Byrnes?CIsidori form for ODE systems with strictly proper transfer function. The ??zero dynamics form?? is exploited to characterize structural properties such as asymptotically stable zero dynamics, minimum phase, and high-gain stabilizability. The zero dynamics are characterized by (A, E, B)-invariant subspaces. Secondly, it is shown that the ??funnel controller?? (that is a static non-linear output error feedback) achieves, for all DAE systems with asymptotically stable zero dynamics and transfer function with proper inverse, tracking of a reference signal by the output signal within a pre-specified funnel. This funnel determines the transient behaviour.     

On a class of optimal abstractions of finite-state machines

Anabstraction A of an fsmM consists in partitioning its states, inputs, and outputs into groups, thus turning it into a non-deterministic fsmM _A. For fixed sets of states, inputs, and outputs, and abstraction generally maps a number of machinesM defined on these sets into the sameM _A. We would like to find anoptimal abstractionA ^* which minimizes this number, while lumping the states, inputs, and outputs into a specified number of classes. We extend these ideas to an fsmM operating in a random environment, and show that the abstraction results in a probabilistic fsm . Thinking of changes inM's output map as resulting in machinesM≠MM, we want to find anA ^* that minimizes the number ofMM which are such that the transition probabilities of their abstracted version are identical to those of the specification machine . SuchMM arise from statistically-undetectable output faults inM. Abstractions are directly applicable to the monitoring of a complex system by an observer for deviations from correct behavior (faults). Complex systems are usually accessible through restricted interfaces, which do not allow the observer to distinguish among all states, inputs, and outputs, thus rendering some faulty transitions undetectable. An optimal interface design will minimize the number of such undetectable faults. Assuming that only single-transition output faults occur inM, we show that each of the classes into which the abstraction lumps the outputs contributes a number of undetectable output faults. We then show that the problem of partitioning the outputs into a given number of classes that minimizes the maximum of these numbers is NP-complete. However, we give (a) an approximate minimization algorithm, running in time linear in the number of classes and quadratic in the number ofM's outputs, and (b) a lower bound on the minimum, computable in the same amount of time. The concept of optimal abstractions is illustrated by numerical results on combinational logic circuits that perform arithmetical operations. The results shed light on the trade-off between model simplification and the ability to detect erroneous behaviors in complex systems.     

Reduced‐order design of high‐order sliding mode control system

To design an rth (r>2) order sliding mode control system, a sliding variable and its derivatives of up to (r ? 1) are in general required for the control implementation. This paper proposes a reduced‐order design algorithm using only the sliding variable and its derivatives of up to (r ? 2) as the extension of the second‐order asymptotic sliding mode control. For a linear time‐invariant continuous‐time system with disturbances, it is found that a high‐order sliding mode can be reached locally and asymptotically by a reduced‐order sliding mode control law if the sum of the system poles is less than the sum of the system zeros. The robust stability of the reduced‐order high‐order sliding mode control system, including the convergence to the high‐order sliding mode and the convergence to the origin is proved by two Lyapunov functions. Simulation results show the effectiveness of the proposed control algorithm. Copyright © 2010 John Wiley & Sons, Ltd.     

On the Gröbner basis triangularization of constraint equations in natural coordinates

Efficient dynamic simulation code is essential in many situations (including hardware-in-the-loop and model-predictive control applications), and highly beneficial in others (such as design optimization, sensitivity analysis, parameter identification, and controller tuning tasks). When the number of modeling coordinates n exceeds the degrees-of-freedom of the system f, as is often the case when closed kinematic chains are present, the governing dynamic equations consist of n second-order ordinary differential equations (ODEs) coupled with m=n?f algebraic constraint equations. This set of n+m index-3 differential-algebraic equations can be difficult to solve in an efficient yet accurate manner. Embedding (or generalized coordinate partitioning) can be used to obtain f ODEs (one for each independent acceleration), which are generally more amenable to numerical integration; however, the dependent positions are typically computed from the independent positions at each time step. Newton–Raphson iteration is often used for solving the position-level kinematics, but only provides solutions to within a specified tolerance, and can require several iterations to converge. In this work, Gröbner bases are used to obtain recursively solvable symbolic solutions for the dependent positions, which can then be evaluated to within machine precision using a fixed number of arithmetic operations. Natural coordinates are particularly attractive in this context, since the resulting constraint equations are maximally quadratic polynomials and are, therefore, easily triangularized. The proposed approach is suitable for use in an automated formulation procedure and, as demonstrated by three examples, is capable of generating highly efficient simulation code with minimal additional effort required at the formulation stage.     

Solving shortest paths efficiently on nearly acyclic directed graphs

Shortest path problems can be solved very efficiently when a directed graph is nearly acyclic. Earlier results defined a graph decomposition, now called the 1-dominator set, which consists of a unique collection of acyclic structures with each single acyclic structure dominated by a single associated trigger vertex. In this framework, a specialised shortest path algorithm only spends delete-min operations on trigger vertices, thereby making the computation of shortest paths through non-trigger vertices easier. A previously presented algorithm computed the 1-dominator set in O(mn) worst-case time, which allowed it to be integrated as part of an O(mn+nrlogr) time all-pairs algorithm. Here m and n respectively denote the number of edges and vertices in the graph, while r denotes the number of trigger vertices. A new algorithm presented in this paper computes the 1-dominator set in just O(m) time. This can be integrated as part of the O(m+rlogr) time spent solving single-source, improving on the value of r obtained by the earlier tree-decomposition single-source algorithm. In addition, a new bidirectional form of 1-dominator set is presented, which further improves the value of r by defining acyclic structures in both directions over edges in the graph. The bidirectional 1-dominator set can similarly be computed in O(m) time and included as part of the O(m+rlogr) time spent computing single-source. This paper also presents a new all-pairs algorithm under the more general framework where r is defined as the size of any predetermined feedback vertex set of the graph, improving the previous all-pairs time complexity from O(mn+nr²) to O(mn+r³).     

On pole assignment of linear systems by gain output feedback

In this paper, the Geometric Approach is used to derive in a straightforward way a sufficient condition for pole assignability by gain output feedback. This result leads to a pole assignment procedure which reduces to solving a system of min (n - m, n - p) polynomial equations where n is the number of states, m the number of inputs, p the number of outputs. In the case where m + p > n, this system clearly appears to be linear. The degrees of freedom related to the pole assignment problem are expressed in terms of (right or left) eigenvectors.     

Minimizing migrations in fair multiprocessor scheduling of persistent tasks

Suppose that we are given n persistent tasks (jobs) that need to be executed in an equitable way on m processors (machines). Each machine is capable of performing one unit of work in each integral time unit and each job may be executed on at most one machine at a time. The schedule needs to specify which job is to be executed on each machine in each time window. The goal is to find a schedule that minimizes job migrations between machines while guaranteeing a fair schedule. We measure the fairness by the drift d defined as the maximum difference between the execution times accumulated by any two jobs. As jobs are persistent we measure the quality of the schedule by the ratio of the number of migrations to time windows. We show a tradeoff between the drift and the number of migrations. Let n = qm + r with 0 < r < m (the problem is trivial for n ≤ m and for r = 0). For any d ≥ 1, we show a schedule that achieves a migration ratio less than r(m − r)/(n(q(d − 1)) + ∊ > 0; namely, it asymptotically requires r(m − r) job migrations every n(q(d − 1) + 1) time windows. We show how to implement the schedule efficiently. We prove that our algorithm is almost optimal by proving a lower bound of r(m − r)/(nqd) on the migration ratio. We also give a more complicated schedule that matches the lower bound for a special case when 2q ≤ d and m = 2r. Our algorithms can be extended to the dynamic case in which jobs enter and leave the system over time.     

A Multi‐sided Bézier Patch with a Simple Control Structure

A new n‐sided surface scheme is presented, that generalizes tensor product Bézier patches. Boundaries and corresponding cross‐derivatives are specified as conventional Bézier surfaces of arbitrary degrees. The surface is defined over a convex polygonal domain; local coordinates are computed from generalized barycentric coordinates; control points are multiplied by weighted, biparametric Bernstein functions. A method for interpolating a middle point is also presented. This Generalized Bézier (GB) patch is based on a new displacement scheme that builds up multi‐sided patches as a combination of a base patch, n displacement patches and an interior patch; this is considered to be an alternative to the Boolean sum concept. The input ribbons may have different degrees, but the final patch representation has a uniform degree. Interior control points—other than those specified by the user—are placed automatically by a special degree elevation algorithm. GB patches connect to adjacent Bézier surfaces with G¹continuity. The control structure is simple and intuitive; the number of control points is proportional to those of quadrilateral control grids. The scheme is introduced through simple examples; suggestions for future work are also discussed.