Polynomial Control Toolbox: A New Aid for Teaching Linear Systems in MATLAB

A new solution to the quadratic optimal tracking problem in linear multivariable discrete-time systems is presented. The approach is based on polynomial input-output models. The optimal control law consists of a plant output feedback and a reference feedforward. It is obtained by performing spectral factorization and then solving two matrix polynomial equations. The design procedure is simple and well suited for systems with inaccessible states.

The simulation problem for high order linear differential systems

This chapter covers two issues (along many others) relating to complex systems control. The main theme is connected with control of Multi-Input Multi-Output (MIMO) systems. If the controlled system has more inputs than outputs (further labelled as MI+MO) there exist many combinations of the inputs for one combination of the outputs. We are able to reach desired system outputs (the main control aim) with many input combinations. This situation is very interesting from practical point of view. Usually optimal inputs combination exists – from some point of view. This combination leads for example to minimum energy consumption, maximum production efficiency or minimum machinery load etc. In practice the set of possible inputs combination is reduced because of constrains and the best feasible combination lies very often on constrain. It would be suitable to extend the controller design to include supplementary demand simultaneously with the fulfilment of the main control aim – to ensure best feasible input combination, too. The common advanced controller like LQ controller has no problem with MIMO system which has different number of inputs and outputs in contrary to standard controllers designed as decentralized control. However the constrains respecting within the standard LQ controller design is not possible. Another advanced controller – Model Predictive Controller (MPC) allows constrains handling (Camacho & Bordons, 1999), (Maciejowski, 2002), (Rossiter, 2003) but the standard controller design doesn’t solve which combination of inputs will occur in the steady-state in case of system with more inputs than outputs. One possibility how to achieve optimal inputs combination is to formulate one term of the cost function connected with inputs penalization as a deviation from ideal inputs combination. This approach is used e.g. in (Novak, 2009) but according to the opinion of authors this approach isn’t as universal as following proposal. We have suggested adding another term into the cost function of predictive controller – terminal state in a form of the deviation from desired terminal state. The desired terminal state is chosen that it corresponds to feasible optimal inputs combination and a value of the set-point at the end of the control horizon. Authors call this technique Steady-state optimization because the influence of the terminal state deviation comes to light namely in steady-state when the main control aim (desired output combination) is or has been already fulfilled see also (Dusek & Honc, 2009). The controller ensures both main and supplementary control aims – achievement of desired outputs and inputs moving to an optimal combination. An incorporation of the terminal state into the cost function has also another advantage. The

Let P be an arbitrary field and let P(x) be a ring of polynomials over P. Denote by Pn and Pn (x) the rings of n xn matrices over P and P(x), respectively. Consider a nonsingular polynomial matrix A(x) e P~ (x) (detA(x) $ 0) and represent it as a matrix polynomial over the field P: is a unital matrix polynomial. In this paper, we establish conditions under which a unital divisor D (x) extracted from the matrix polynomial A (x) is determined uniquely by its characteristic polynomial detA (x) = d(x). The re- sult obtained enables one to indicate the class of matrix polynomials for which this unital divisor is unique for a given characteristic polynomial. We also show how this result can be used when solving matrix polynomial equations. The uniqueness of a tmital divisor for a matrix polynomial over an algebraically closed field with char- acteristic zero was studied in ( 1 ). It is known (2) that, for a matrix A (x) e Pn (x), there exist matrices U(x), V(x) ~ GL n (Pn (x)) such that U (x)A (x) V(x) = F A (X) = diag (al(x), a2(x) ..... an(x)), where aj(x) e P(x), j= 1, 2 ..... n, are unital polynomials and ai(x ) (ai+l(x), i = 1, 2 ..... n - 1. The matrix FA(X ) is called a canonical diagonal form of the matrix A(x). It is also known that if a matrix is nonsingular, then it can be reduced to the normal Hermite form by right elementary transformations, i.e., for the matrix A (x), there exists a matrix W(x) e GLn (P(x)) such that

In §21, we discovered (c.f. Theorem 21.9) that a real m-input, p -output time-invariant continuous- or discrete-time input-output linear system with impulse response t → H(t) or k → H(k) was realizable if and only if its transfer function s → G (s) or z → G (z) agreed on its region of convergence ROC(H) with a real proper (p × m) matrix rational function of s or of z. The constant realizations for such a system was shown in Facts 21.10 to lie in one-to-one correspondence with matrix quadruples (A, B, C, D) which satisfied $$\begin{aligned} & & & G\left( s \right) = C{\left( {s{I_n} - A} \right)^{ - 1}}B + D,\quad s \in ROC\left( H \right), \\ or \\ G\left( z \right) = C{\left( {z{I_n} - A} \right)^{ - 1}}B + D,\quad z \in ROC\left( H \right). \\ \end{aligned} $$

Let [Ctilde](s)[Dtilde](s)−1 be a right coprime matrix fraction description of the (r × m) strictly proper transfer function matrix T(s) of a linear multivariable system Σ = (A, B, C). In this paper we establish a direct relation between (i) the algebraic concept of a right divisor C R(s) of the ‘numerator’ matrix [Ctilde](s), i.e. a (not necessarily square) polynomial matrix C L(s) that satisfies: [Ctilde](s) = C L(s)C R(s) for some polynomial matrix C L(s), and (ii) the geometric concept of an (A, B)-invariant subspace which is contained in the kernel of C. A special class of right divisors C R(s) of C(s) which correspond to (A, B)-invariant subspaces in ker C is characterized by a number of properties and a simple formula is presented which expresses the above correspondence. By generalizing the concept of a greatest right divisor (GRD) of a polynomial matrix to include also possibly non-square polynomial matrices, it is shown that GRDs of the numerator matrix C(s) correspond to the maximal (A...

Importance of Polynomial in Coefficient Diagram Method

A collection of algorithms developed in a computer algebra package (MapleV) using polynomial matrix theory is described. The developed algorithms provide a medium in which polynomial matrix operations are carried out. Most importantly, these polynomial matrix procedures enable the design and analysis of multivariable control systems using the algebraic or polynomial equation approach. This algebraic design would have been extremely difficult to carry out in a strict numeric computing environment. The use of MapleV has provided symbolic results quickly and efficiently, with a tremendous gain in time and with minimal effort.

Polynomial Equations Giving a Proper Feedback Compensator for a Strictly Proper Plant

The purpose of this paper is to give four new applications of the Quillen-Suslin theorem to mathematical systems theory. Using a constructive version of the Quillen-Suslin theorem, also known as Serre's conjecture, we show how to effectively compute flat outputs and injective parametrizations of flat multidimensional linear systems. We prove that a flat multidimensional linear system is algebraically equivalent to the controllable 1-D dimensional linear systems obtained by setting all but one functional operator to zero in the polynomial matrix defining the system. In particular, we show that a flat ordinary differential time-delay linear system is algebraically equivalent to the corresponding ordinary differential system without delay, i.e., the controllable ordinary differential linear system obtained by setting all the delay amplitudes to zero. We also give a constructive proof of a generalization of Serre's conjecture known as Lin-Bose's conjecture. Moreover, we show how to constructively compute (weakly) left-/right-/doubly coprime factorizations of rational transfer matrices over a commutative polynomial ring. The Quillen-Suslin theorem also plays a central part in the so-called decomposition problem of linear functional systems studied in the literature of symbolic computation. In particular, we show how the basis computation of certain free modules, coming from projectors of the endomorphism ring of the module associated with the system, allows us to obtain unimodular matrices which transform the system matrix into an equivalent block-triangular or a block-diagonal form. Finally, we demonstrate the package QuillenSuslin which, to our knowledge, contains the first implementation of the Quillen-Suslin theorem in a computer algebra system as well as the different algorithms developed in the paper.

Three existing numerical procedures for solution of a linear matrix polynomial equation AX+BY=C have been tested, evaluated and compared in experimental manner. Thousands of examples have been computed thus far covering both generic (randomly created) data and ill-conditioned cases (with 'almost common factors'). The most successful method in the triple appears to be the one based on interpolation (PIM). It has been able to handle generic examples with the polynomial matrices A, B up to sizes 20/spl times/20 of degree 5 as well as ill-conditioned cases with 5/spl times/5 matrices of degree 5 suffering with almost common factors different from the precise ones (for which the equation becomes unsolvable) relatively only by 2/sup -30/. The other two methods considered based on elementary operations and state space realization were found to exhibit some drawbacks in their present form. >

We consider the adaptive tracking problem for a chain of integrators, where the uncertainty is static and functional. The uncertainty is specified by $L^2/L^\infty$ or weighted $L^2/L^\infty$ norm bounds. We analyse a standard Lyapunov-based adaptive design which utilises a function approximator to induce a parametric uncertainty, on which the adaptive design is completed. Performance is measured by a modified LQ cost functional, penalising both the tracking error and the control effort. With such a cost functional, it is shown that a standard control design has divergent performance when the resolution of a mono-resolution approximator is increased. The class of mono-resolution approximators includes models popular in applications. A general construction of a class of approximators and their associated controllers which have a uniformly bounded performance independent of the resolution of the approximator is given.

In this paper the rate of stability of solutions of matrix polynomial equations of the typeA0+A1X+A2X2+...+A m X m =0 is studied. Particular attention is given to the case where the matrix polynomialL(λ):=A0+A1λ+A2λ2+...+A m λ m is weakly hyperbolic, i.e., for every non-zero vectorx the scalar polynomial 〈L(λ)x, x〉 has only real roots. Also the rate of stability of solutions of matrix quadratic equations of the typeXBX+XA-DX-C=0 is studied. Here the special case that is of interest to continuous-time optimal control theory, that is, the case whereB=B* is positive semidefinite andC=C*,A=−D*, is discussed in detail. The analogous theory for the discrete-time optimal control leads to the equation $$X = A^* XA + Q - (B^* XA)^* (R + B^* XB)^{ - 1} B^* XA,$$ and the rate of stability of solutions of this equation is also studied. Most of the problems are discussed in both real and complex settings.

Polynomial matrices play an important part in linear system calculations. New computational procedures are given for calculation of the Smith normal form and the greatest common right divisor of polynomial matrices. It is shown how suitable transformation matrices can be determined for the calculation of the Smith normal form, and how a set of polynomial matrix multipliers can be calculated for the greatest common right divisor problem. Neither of these algorithms relies on explicit calculation of the greate3t common divisor of polynomials. Limited numerical experience has shown that the3e algorithms are both fast and accurate.

Necessary and Sufficient Conditions for Delay-Dependent Asymptotic Stability of Linear Discrete Time Delay Autonomous Systems