Uncertain Models and Robust Control

The introductory survey is intended to provide the reader with the aims of uncertain models and structures of uncertainties. An overview covering several robust control concepts is embedded into general techniques of how to overcome uncertain systems. Furthermore, concepts are included that already exist and that are likely to be applicable in the future.

Norms are widespread in the techniques of uncertain modelling and robust control. Multivariable systems and the state-space representation require valuating vectors and matrices by a single number. Most of the important relations for norms are listed or derived in this chapter. This material may be considered as a kind of reference, and the chapter may be skipped when the reader is already familiar with this subject.

Consider a function f with the value f(t) at a certain instant t and assume that the integral of the pth power of the absolute value |f(t)| exists within a region R. Then, this function f is said to be within the function space (or set of functions) L_p(R). In most cases, R is given by an interval [t_o, ∞). The function norm ∥If∥_p of the entire function f is defined by 3.1$${\left\| f \right\|_p} \buildrel \Delta \over = {\left[ {\int_R {{{\left| {f(t)} \right|}^p}} dt} \right]^{1/p}}{\rm{ }}\forall p \in \left[ {1,\infty )} \right.$$

In sensitivity analysis and in system analysis there is a frequent need for Kronecker products. The Kronecker product of two matrices (or direct product, tensor product) is defined by a partitioned matrix whose (i,j)-partition is A_ijB4.1$${\rm{A}} \otimes {\rm{B}} \buildrel \Delta \over = \left( {\begin{array}{*{20}{c}} {{A_{11}}{\rm{B}}}&{{A_{12}}{\rm{B}}}& \ldots &{{A_{1n}}{\rm{B}}}\\ {{A_{21}}{\rm{B}}}&{{A_{22}}{\rm{B}}}& \ldots & \vdots \\ \vdots & \vdots & \ddots & \vdots \\ {{A_{m1}}{\rm{B}}}&{{A_{m2}}{\rm{B}}}& \ldots &{{A_{mn}}{\rm{B}}} \end{array}} \right) = {\rm{matrix}}\left[ {{{\rm{A}}_{ij}}{\rm{B}}} \right],{\rm{ }}\begin{array}{*{20}{c}} {{\rm{A}} \in {C^{nxm}}}\\ {{\rm{B}} \in {C^{rxs}}}\\ {{\rm{A}} \otimes {\rm{B}} \in {C^{nrxms}}} \end{array}$$

In control theory, employing vectors and matrices, formulas become short and simple. When studying dynamic behaviour and its parameter sensitivity, matrix calculus frequently is applied. In these cases, the differentiation of scalar-, vector- and matrix- valued functions with respect to scalars, vectors or matrices is required. An overview of the nine cases arising is given in Table 5.1. Moreover, Kronecker algebra as discussed in former chapters provides shorter and more concise calculations.

Differential sensitivity of an eigenvalue is given by the quotient of an infinitesimal change of the eigenvalue λ[F] and an infinitesimal change of a matrix K on which the eigenvalue depends when F = A + BKC . Small-scale robustness is obtained when the differential sensitivity is smalI. A geometrie interpretation of the differential sensitivity of an eigenvalue with respect to a matrix is given in Eq.(32.91).

Referring to Eq.(4.39) let M be the (n,n)-matrix A of the open-loop or closed-loop coefficients 7.1$${e^{{{\rm{I}}_m} \otimes {\rm{A}}t}} = \sum\limits_{k = 1}^n {{{\rm{I}}_m}} \otimes \left( {{a_k}a_k^{ \triangleleft T}} \right){e^{{\lambda _k}\left[ A \right]t}}$$ where I_m is the identity matrix with arbitrary dimension m x m, a_k is the kth (right) eigenvector of A associated with eigenvalue γ_k[A]. Furthermore, aΔ_k is the left eigenvector or the right eigenvector of AT, normalized with a_k (i.e., aΔT_kai = δ_ik).

Consider the dynamic system with output feedback controller as depicted in Fig. 8.1 8.1$${\rm{\dot x(}}t) = {\rm{Ax(}}t) + {\rm{Bu(}}t){\rm{ x}} \in {{\rm{R}}^n}$$8.2$${\rm{y(}}t) = {\rm{Cx(}}t){\rm{ x}} \in {{\rm{R}}^r}$$8.3$${\rm{u(}}t) = {\rm{Ky(}}t) + {\rm{ x}} \in {{\rm{R}}^r}{y_{_{ref}}}(t){\rm{ u,}}{{\rm{y}}_{ref}} \in {R^m}$$

This chapter is devoted to the following topics: •Riccati controller and Riccati matrix differential sensitivity with respect to changes in parameters, supposing that the system matrix and the input matrix depend on these parameters.•Differential sensitivity of an index of performance with respect to parameters contained in the system and input matrix. When I should be minimized with respect to p, the zeros of ∂I/∂p are calculated. In order to apply Newton-Raphson algorithm the second derivative ∂2I / ∂pT∂p is determined.•Reducing performance sensitivity can be achieved via gradient methods, augmenting the index of performance by differential sensitivity expressions.•Control actions softer than optimal control are obtained when a prespecified index of performance is approached via gradient methods.

Consider the plant 1$${\rm{\dot x(t) = A(p)x(t) + B(p)u(}}t){\rm{B}} \in {R^{nxm}},{\rm{p}} \in {{\rm{R}}^{{n_p}}}$$ where A(p) and B(p) are matrix-valued functions of a slowly varying parameter vector p. Hence, x(t) depends on p . The subscript 0 denotes the nominal values. Assume that a quadratic performance has to be minimized 2$$I = \int_0^\infty {[{x^T}(t)Qx(t) + {{\rm{u}}^T}} (t){\rm{Ru}}(t)]dt$$ and the optimal control variable u(t) is 3$${{\rm{u}}^ \star }(t) = Kx(t) = - {R^{ - 1}}{B_0}Px(t)$$

Solving the linear first-order time-varying differential equation the 11.1$$\dot x(t) = a(t)x(t) + u(t)$$ homogeneous differential equation 11.2$$\dot x(t) = a(t)x(t){\rm{ or }}\frac{{\dot x(t)}}{{x(t)}} = \frac{d}{{dt}}\ln x(t) = a(t)$$ is considered first. 11.3$$In{\rm{ }}x(t) = \int_{{t_o}}^t {a(\tau } ) + \ln {\rm{ }}k$$11.4$$x(t) = k{\rm{ exp }}\int_{{t_o}}^t {a(\tau )d\tau \underline{\underline \Delta } } k\varphi (t,{t_o})$$

Stability results in interval polynomials and interval matrices can be applied to solve the problem of robust control in spite of the fact that there is no simple relation between the tolerances or the uncertainty of the plant and the interval matrix or interval characteristic polynomial of the closed-loop control system. A property is said true with respect to the interval matrix A_I if the property holds for every A ∈ A_I.

The matrix U denotes a positive matrix if each entry U_ij > 0 ∀i, j . A matrix is termed positive definite if it has properties in what is to follow. A square matrix Q is positive definite if 13.1$${{\rm{x}}^T}Qx > 0{\rm{ }}\forall {\rm{x}} \ne {\rm{0}}$$ and positive semidefinite (non-negative definite) if 13.2$${{\rm{x}}^T}Qx \ge 0{\rm{ }}\forall {\rm{x}}$$

This chapter is devoted to the time-domain approach of stability robustness of discretetime systems. Uncertainty frequently occurs associated with the parameters of the process. However, with regard to varying random sampling another type of uncertainty may arise in control systems. The stability margin and the expectation of the stability margin can be assessed by simple formulae (Jury, E.I., and Tsypkin, Y.Z., 1971; Weinmann, A., 1981 and 1985).

Pole assignment is a well-known method for controller design. If the plant parameter vector is characterized by some uncertainty and if there is some flexibility in the desired closed-Ioop pole vector, the problem of finding a robust controller arises. The solution of this problem is not a simple one of solving linear algebraic equations as in the case of plants given in companion form. The problem is of higher complexity and requires an iterative algorithm, e.g., following a steepest descent method.

This chapter is devoted to several models based on least squares theory and system component interconnection. Optimal and suboptimal solutions and associated suboptimal controllers are considered. Solving the problems, multistage computational structure is employed. Large-scale interconnected systems for practical engineering purposes require decentralized controllers together with a suitable design procedure.

The ellipsoidal set-theoretic approach is used to formulate the control problem arising when both input and system state matrix are perturbed. The set of perturbed parameters is bounded in a given compact convex set. The objective is to find a linear state-feedback controller K such that both the system state and control variables are minimally bounded by an ellipsoid and the ellipsoids satisfy a minimal performance functional. The boundary ellipsoids are minimized in a mean size given by the length sum of semi-major axes. The ellipsoidal set-theoretic approach originally was documented by Usoro, P.B., et al. 1982. The design method in the following was presented by Wang, S.D., and Kuo, T.S., 1990.

Consider a multivariable dynamic system of order n with input u(t) ∈ Rm and output y(t) ∈ Rr. The system state variable x(t) is an n-vector. Observability and controllability is preassumed. UsuaIly x(t) is not available. If the output y(t) does not provide sufficient information for high-performance control, observers are implemented to estimate either the entire vector or apart of it. In the presence of process noise and/or measurement noise observers are replaced by KaIman filters.

Overshoot and speed requirements of a control system are important performance objectives. There is a strong need for balancing stability robustness versus the performance. This short chapter is devoted to optimizing stability robustness and performance as cited above. The Overshoot behaviour of a closed-loop system is defined as the maximum value of the spectral norm of its transition matrix Φ_cl (t) weighted by some weighting matrices W₁, W₂19.1$${M_{\max }} = \mathop {\max }\limits_t {\left\| {{{\rm{W}}_1}{\Phi _{cl}}(t){{\rm{W}}_2}} \right\|_s}{\rm{ where }}{\Phi _{cl}}(t) = {e^{{\rm{F}}t}}$$

Stability and stability robustness are important properties of control systems. Hence, particular attention is paid to this topic. However, control system performance, disturbance rejection and tracking facilities (Jayasuriya, S., 1987) may not be neglected. Function spaces and Ln_p-norms yield a very useful tool both for investigating stability and disturbance rejection and tracking problems in the face of uncertainties.

Let the set U be the closed set of values s which are not located in a desired stability region Г, e.g., U is the closed right-half plane. The polynomial a( s, Q) is denoted U-stable if and only if no root of a(s, Q) is located in the undesired region, ie., 21.1$$a(s,Q) = \{ a(s,q){\rm{ : q}} \in Q\} \ne 0{\rm{ }}\forall s \in U$$

Real matrices are matrices with elements of real numbers or functionsj they are often applied in the state-space approach of control systems, various regression techniques, convolution sums but also in some other topics related to the time domain, e.g. stability radius techniques. If design problems of multi variable systems are treated in the s-plane or in the frequency domain use is made of transfer matrices with entries of complex numbers and functions. To avoid confusion, symbols of a different kind are chosen in most cases, e.g., G for a complex matrix and A for areal one.

Consider the homogeneous equation of a dynamic system $${\rm{\dot x(}}t) = {\rm{Ax(}}t){\rm{ }} \in {{\rm{R}}^n}$$ where A represents the nominal system which is assumed stable. Let the system be linearly perturbed by an additive error matrix ΔA 23.1$${\rm{\dot x(}}t) = ({\rm{A + }}\Delta {\rm{A)x(}}t)$$ where ΔA is spectral norm bounded, i.e. ∥A∥_s < c. Martin, J.M., 1987 proposed a measure m_M for the stability robustness of the linear state-space system model 23.2$$[{m_M}\underline{\underline \Delta } \mathop {\min }\limits_{w \ge 0} {\frac{1}{{\left\| {{\rm{L(}}w)} \right\|}}_s} = {\frac{1}{{{{\max }_{w \ge 0}}\left\| {{\rm{L(}}w)} \right\|}}_s}$$23.3$$where L(\omega ) \triangleq {{(j\omega I - A)}^{{ - 1}}}.$$

Plant uncertainties and plant model variations must be taken into consideration in almost every application. With regard to plant uncertainties the global control system performance decreases. The subject of this chapter is the application of singular values to the overview problem of balancing system stability robustness, system performance and plant uncertainty.

Analysis and design of control systems in the frequency domain is one of the best known methods in control engineering. The stability of single-input single-output systems is investigated via transfer function calculus and Nyquist criterion (Leithead, W.E., and O'Reilly, J., 1991). Multi-input multi-output systems require the transfer matrix operations and characteristic loci. When plants are perturbed characteristic loci can be still used if they are augmented by E-contours or structured E-contours.

Control theory is characterized by a tradeoff between the level of control performance and the tolerance with respect to plant uncertainties. The structured singular value (Doyle, J.G., 1982; Doyle, J.G., et al. 1982) provides a reliable measure to satisfy both performance and robustness requirements. The structured singular value can be considered as a measure generalizing those methods which are based on the ordinary-singular-value decomposition.

Robustness is usually considered as stability robustness in the face of perturbed plant dynamics. If, additionally, a certain level of performance has to be guaranteed, robustness is modified to performance robustness. In the s-domain the method of maximum singular values and structured singular values can easily be reformulated to obtain acceptable performance in the presence of plant uncertainties.

Consider a multivariable system as depicted in Fig. 28.1. The controller is decomposed into three transfer matrices K_r(s), K_f(s) and K_y(s) . The plant G(s) is characterized by an uncertainty in parallel △G(s). The uncertainty is caused by unknown plant structure and parameters or by reduced-order model approximation of the plant.

In this chapter conditions are derived such that asymptotic tracking and output regulation (or disturbance rejection) occurs in multivariable systems. The system in considered robust with respect to plant parameter perturbations as long as they do not affect system stability and the asymptotic behaviour mentioned above is guaranteed. The reference input and the disturbance are assumed to satisfy a certain differential equation. This problem is also known as the robust servomechanism problem. A servo-compensator is incorporated which consists of r unstable compensators with identical dynamics, corresponding to the class of reference/disturbance. The integer r is the dimension of the output signal y(t).

The factorization of rational matrix-valued functions of the complex variable s is frequently employed in designing robust controllers. Various factorization algorithms developed, e.g., by Youla, D.C., 1961,. Youla, D.C., et al. 1976; Francis, B.A., 1987,. Safonov, M.G., and Verma, M.S., 1985,. Vidyasagar, M., and Kimura, H., 1986 are presented in this chapter.

In Fig. 31.1 a single-input single-output system is given. A robust controller K(s) should be designed to guarantee sufficient closed-loop performance even in the presence of plant uncertainties. The actual plant transfer function is G_p = {1 + ΔN[u(t),t]}G where G denotes the nominal plant and ΔN[u(t), t] is a real-valued nonlinear time-varying function of the input signal u(t). The uncertainty ΔN is considered bounded by the sector gain γN 31.1$$\Delta N\left[ {u(t),t} \right] \le \gamma N\left| {u(t)} \right|{\rm{ }}\forall u(t){\rm{ where }}\Delta N\left[ {0,t} \right] = 0{\rm{ }}\forall t$$ The nonlinearity is not essential in this chapter but the derivations can easily be extended to several types of nonlinear uncertainty.

Sliding motions in a variable structure control system yield a design method which is applicable to uncertain dynamical systems. A variable structure control system has the central feature of switching on one or more manifolds in the state space. When acting in the sliding motion, the system state continuously intersects the switching hyperplane in different directions. If a system switches on all the switching manifolds together, the system is said to be in the sliding mode (Dorling, C.M., and Zinober, A.S.I., 1986). The sliding mode is obtained by a discontinuous control action. The system changes its structure when crossing the switching surface. Systems operating in sliding mode are therefore denoted variable structure systems.

Models of dynamic systems are frequently lacking in sufficient knowledge of several highfrequency parameters, is spite of strength in identification. Hence, those parts of the system incorporating inaccessible high-frequency dynamics are referred to as parasitic high-frequency dynamics. Neglecting high-frequency dynamics seems comfortable on the one hand since the system is of reduced order but on the other hand, the resulting closedloop control may have poor stability properties.

Modelling of large-scale dynamic systems requires differential equations of high dimension. Control algorithms can often only be derived and implemented for models of reduced dimension. In view of this, the concept of aggregation provides simplified representation and is suitable for approximate synthesis. The relation z = Γx with constant matrix r is used to transform the original model or plant state vector x into the aggregated system with the aggregated state vector z . The objective is that z contains the significant portion of the plant dynamics, see Fig. 34.1. Then, one has original plant: 34.1$${\rm{\dot x = Ax + Bu x}} \in {{\rm{R}}^n},{\rm{ u}} \in {{\rm{R}}^m}$$ aggregation: 34.2$${\rm{z = }}\Gamma {\rm{x, }}\Gamma \in {{\rm{R}}^{lxn}}$$ aggregated model: 34.3$$\begin{array}{*{20}{c}} {\dot{z} = Fz + Gu} & {z \in {{\mathcal{R}}^{l}}.} \\ \end{array}$$

Consider the matrix set X = {X_o X₁ X₂ ... X_q-1 and a mapping T_t, given by the matrix sum weighted with powers of time t35$${T_t}X \buildrel \Delta \over = \sum\limits_{k = 0}^{q - 1} {{{\rm{X}}_k}{t^k}}$$

Classical orthogonal polynomials, generalized and shifted orthogonal polynomials serve as an excellent tool for modelling plants and processes in an approximative way. Moreover, solutions via orthogonal polynomials give a straightforward algorithm in reducing various dynamic problems to the solution of a set of algebraic linear equations. The integral of a function with respect to time is replaced by premultiplying the vector of the decomposed signal by an operational matrix. Furthermore, the product of functions and even timevarying coefficients and nonlinear relations are reduced to algebraic manipulations. Among these problems are system analysis, variational calculus, optimal control, identification and model reduction. Orthogonal functions have been applied in control measurement and correlation techniques for a long period, see e.g. Kitamori, T., 1960; Douce, J.L., and Roberts, P.D., 1964. The classical Fourier series expansion, truncated to a certain amount of terms, can also be applied to solve various dynamic problems (Paraskevopoulos, P.N., et al. 1985; Cheok, K.C., et al. 1989).

For obvious reasons, approximating signals by piecewise constant or piecewise linear signals is a very simple approach. Walsh functions or block-pulse functions are applied in order to obtain concise results. Piecewise constant signals provide simple algorithms for approximating signals and processes.

The orthogonal component coefficient vector x⌝ is given by Eq.(36.16). Thus, x(t) is decomposed into k terms. Starting now with the orthogonal expansion coefficient vector associated with (t) and using h_o(t) = 1, the original function x(t) can be expressed as follows 38.1$$x(t) = \int_o^t {\dot x(t)dt + x(0) = \int_o^t {{{\dot x}^{\neg T}}} } hdt + x(0) = {\dot x^{\neg T}}\int_o^t {h(t)dt + x(0)}$$38.2$$x(t) = {\dot x^{\neg T}}{{\rm{P}}_M}{\rm{h(}}t) + x(0){h_o}(t) = {\dot x^{\neg T}}{{\rm{P}}_M}{\rm{h + x}}_o^Th;{\rm{ }}{{\rm{x}}_o} \buildrel \Delta \over = {\left[ {x(0){\rm{ 0 0}}...} \right]^T}$$