Generalized Ratio Control of Discrete-Time Systems Generalized Ratio Control of Discrete-Time Systems

This chapter exposes the important connection between ratio control and the state control reflecting equality constraint for linear discrete-time systems, which allows significant reduction in computational complexity and efforts. Based on an enhanced bounded real lemma form, to outperform known approaches, the existence of the state feedback for such defined singular task is proven, and the design procedure based on the linear matrix inequalities is provided. The proposed principle, guaranteeing feasibil-ity of the set of inequalities, improves steady-state accuracy of the ratio control and essentially reduces the design effort. The approach is illustrated on simulation examples, where the validity of the proposed method is demonstrated.


Introduction
The problem of the ratio feedback control is one of the specific topics in the theory of control synthesis. It is well practically motivated by applied realizations but not favorable developed in a state control technique or in combination with the state estimation theory. However, a considerable number of problems in the ratio control design have to deal with systems subjected to constraint conditions, which are other than linear, or directly formulated as singular constrained tasks. In the typical case [1,2] where the system state reflects certain physical entities, constraints usually prescribe the system state, the region of technological conditions. If the ratio control is not formulated as a task with the equality constraints, the application requires further procedures of controlling the evolution of the set-valued ratio. Notably, a special form of the problems can be defined while the system state variables satisfy constraints and interpreted as descriptor systems [3][4][5][6]; but, the system with state equality constraints generally does not satisfy the conditions under which the results of descriptor systems can be used. Moreover, if the design task is interpreted as a singular problem, constraint associated methods have to be developed to design the controller.
In principle, it is possible to design the controller that stabilizes a system and simultaneously forces its closed-loop properties to satisfy given constraints [7,8]. Following the idea of linear quadratic (LQ) control application, these approaches heavily rely on set-valued calculus as well as on min-max theory [9,10], which are not simple and lead to rather cumbersome technical and numerical procedures. A more simple technique, using equality constraints formulation for discrete-time multiinput/multioutput (MIMO) systems, is introduced in Refs. [11,12]. Based on the eigenstructure assignment principle, a slight modification of equality constraint technique is presented in Ref. [13].
Many tasks that arise in state-feedback control formulation can be reduced to standard convex problems that involve matrix inequalities. Generally, optimal solutions of such problems can be computed by using the interior point method [14], which converges in polynomial time with respect to the problem size. A review of the progress made in this field can be found in Refs. [15][16][17] and the references therein. In the given sense, the stability conditions are expressed in terms of linear matrix inequalities (LMI), which have a notable practical interest due to the existence of numerical LMI solvers [18,19].
The chapter devotes the design conditions to obtain a closed-loop system in which minimally two state variables are rebind by the prescribed ratio. The generalized ratio control principle is reformulated as the full-state feedback control with one equality constraint. Solving this problem, the technique for an enhanced BRL representation [20,21] is exploited, to circumvent potentially ill-conditioned singular task concerning the discrete-time systems control design with state equality constraints [22]. Due to application of the enhanced BRL, which decouple the Lyapunov matrix and the system matrices, the design task stays well-conditioned. These conditions impose such control that assures asymptotic stability for time-invariant discrete control under defined equality constraints. The presented way, based on projecting the target state variables into a subset of the system state space, adapts the idea of performing the LQ control principle in the fault tolerant control and the constraint control of discrete-time stochastic systems [23,24].
The outline of this chapter is as follows. Continuing the introduction outlines in Section 1, the problem formulation is principally presented in Section 2. Section 3 is dedicated to the mathematical backgrounds supporting the problem solution and the exploited discrete-time LMI modifications are given in Section 4. These results are used in Section 5 to examine the linearization problems in bilinear matrix inequalities, so that in Section 5, these results can be given with convex formulation of control design condition, guaranteeing a feasible solution of the generally singular design task. Subsequently, numerical examples to illustrate basic properties of the proposed method are presented in Section 6, and Section 7 is finally devoted to a brief concluding remarks.
Throughout the chapter, the following notations are used: x T and X T denote the transpose of the vector x and matrix X, respectively, for a square matrix X < 0 that X is a symmetric negative definite matrix, the symbol I n represents the nth order unit matrix, Y ⊝1 denotes the Moore-Penrose pseudoinverse of a nonsquare Y, ∥ Á ∥ represents the Euclidean norm for vectors and the spectral norm for matrices, I R denotes the set of real numbers and I R n × r the set of all n × r real matrices.

Problem formulation
Through this chapter, the task is concerned with design of the full-state feedback control to discrete-time linear dynamic systems in such a way that the closed-loop system state variables are constrained in the prescribed ratio. The systems are defined by the set of state equations where q(i) ∈ I R n is the vector of the state variables, u(i) ∈ I R r is the vector of the input variables, y(i) ∈ I R m is the vector of the output variables, and nominal system matrices F ∈ I R n × n , G ∈ I R n × r , and C ∈ I R m × n are real matrices, and i ∈ Z + .
The discrete transfer function matrix of dimension m × r, associated with the system Eqs. (1) and (2) is defined as where I n ∈ I R n × n is the identity matrix, ỹ(z) and ũ(z) stand for the Z transform of m dimensional output vector and r dimensional input vector, respectively, and a complex number z is the transform variable of the Z transform [25].
In practice, the ratio control maintains the relationship between two state variables [26,27] and is defined for all i ∈ Z as Assuming the parameter vector e h , the task can be expressed by using the system state vector q(i + 1) as It is evident that the generalized ratio control can be defined by a composed structure of e, as well as by a structured matrix E [28].
The task formulated above means the design problem that can be generally defined as the stable closed-loop system synthesis using the linear full-state feedback controller of the form where K ∈ I R r × n is the controller feedback gain matrix, and the design constraint is considered in the general matrix equality form with E ∈ I R p × n , rank E = p ≤ r. In general, the matrix E reflects prescribed fixed ratio of two or more state variables. The equality Eq. (9) evidently implies ΛEq(i + 1) = 0, where Λ ∈ I R p × p is an arbitrary matrix.
It is considered in the following the discrete-time system is controllable and observable that is, rank zI À F, G ð Þ¼n ∀z ∈ C and rank zI À F T , C T À Á ¼ n ∀z ∈ C, respectively [29], and that all state variables are measurable.

Basic preliminaries
Then all solution to Θ means that where while A ⊝ 1 is the left Moore-Penrose pseudoinverse of A, B ⊝ 1 is the right Moore-Penrose pseudoinverse of B and Θ°is an arbitrary matrix of appropriate dimension.
Proof. (see, e.g., Ref. [15]) Proposition 2. Let Ξ ∈ I R n × n is a real square matrix with nonrepeated eigenvalues, satisfying the equality constraint then one from its eigenvalues is zero, and the (normalized) vector e T is the left raw eigenvector of Ξ associated with the zero eigenvalue.
Proof. If Ξ ∈ I R n × n is a real square matrix satisfying the above given eigenvalues limitation, then the eigenvalue decomposition of Ξ takes the following form where n l is the right eigenvector and m T l is the left eigenvector associated with the eigenvalue z l of Ξ, and {z l , l = 1, 2,…n} is the set of the eigenvalues of Ξ. Then Eq. (13) can be rewritten as follows: If e T ¼ m T h , then orthogonal property Eq. (15) implies and it is evident that Eq. (17) can be satisfied only if z h = 0. This concludes the proof. □ Proposition 3. (Quadratic performance) Given a stable system of the structure Eqs. (1) and (2), then it yields where γ ∞ ∈ I R is the H ∞ norm of the transfer function matrix of the system Eq. (3).
then, evidently, where ∥ H(z) ∥ is H 2 norm of the discrete transfer function matrix H(z).
Since the H ∞ norm property states using the notation ∥ H(z) ∥ ∞ = γ ∞ , then Eq. (21) can be naturally rewritten as Thus, based on the Parseval's theorem, Eq. (22) gives and using squares of the elements, the inequality Eq. (23) subsequently results in Thus, Eq. (24) implies Eq. (18). This concludes the proof. □ If it is not in contradiction with other design constraints, Eq. (18) can be used as the extension to a Lyapunov function candidate for linear discrete-time systems, since it is positive.

Quadratic performances
The above presented assumptions are imposed to obtain LMI structures exploiting H ∞ norm, known as the bounded real lemma LMIs. To simplify proofs of theorems in following, proof sketches of the BRL are presented, since more versions of BRL can be constructed.  1) and (2) is stable with the quadratic performance γ ∞ , if there exist a symmetric positive definite matrix P ∈ I R n × n and a positive scalar γ ∞ ∈ I R such that where I r ∈ I R r × r and I m ∈ I R m × m are identity matrices, respectively.
Hereafter, * denotes the symmetric item in a symmetric matrix.
Proof. (compare, e.g., Refs. [16] and [23]) Defining the Lyapunov function candidate as follows: and, using the description of the state system Eqs. (1) and (2), the inequality Eq. (28) becomes Thus, introducing the notation where Since, using the Schur complement property with respect to the matrix element γ À1 ∞ C T C, Eq. (32) can be rewritten as then, applying the dual Schur complement property, Eq. (33) implies Eq. (26). This concludes the proof. □ Direct application of the second Lyapunov method [30,31] and BRL in the structure given by Eqs. (25) and (26) for affine uncertain systems as well as in constrained control design is in general ill-conditioned owing to singular design conditions [13]. To circumvent this problem, an enhanced LMI representation of BRL is proposed, where design condition proof is based on another form of LMIs.
Proposition 5. (Enhanced LMI representation of BRL) The autonomous system Eqs. (1) and (2) is stable with the quadratic performance γ ∞ , if there exist a symmetric positive definite matrix P ∈ I R n × n , a regular square matrix Q ∈ I R n × n , and a positive scalar γ ∞ ∈ I R such that where I r ∈ I R r × r and I m ∈ I R m × m are identity matrices. Proof. Since, Eq. (1) can be rewritten as with an arbitrary square matrix Q ∈ I R n × n , it yields Now, not substituting Eq. (1) into Eq. (28), but adding Eq. (37) and its transposition to Eq. (28), it can be obtained that Thus, considering Eq. (2), then Eq. (38) can be rewritten as Since Eq. (41) can be written as then, using the dual Schur complement property, Eq. (43) can be transformed in the form To obtain a LMI structure visually comparable with Eq. (26), the following block permutation matrix is defined It is evident that Lyapunov matrix P is separated from the matrix parameters of the system F, G, and C, i.e., there are no terms containing the product of P and any of them. By introducing the slack variable matrix Q, the product forms are relaxed to new products QF and QG, where Q needs not be symmetric and positive definite. This enables a robust BRL, which can be obtained to deal with linear systems with parametric uncertainties, as well as with singular system matrices.
Considering a symmetric positive definite matrix Q ∈ I R n × n , the following symmetric enhanced LMI representation of BRL is evidently obtained.

Corollary 1. (Enhanced symmetric LMI representation of BRL)
The autonomous system Eqs. (1) and (2) is stable with the quadratic performance γ ∞ , if there exist symmetric positive definite matrices P, Q ∈ I R n × n and a positive scalar γ ∞ ∈ I R such that where I r ∈ I R r × r , I m ∈ I R m × m are identity matrices.
Note, Corollary 1 provides the identical condition of existence to Proposition 4, if the equality P = Q is set.

Control law parameter design
The state-feedback control problem is finding, for an optimized (or prescribed) scalar γ > 0, the state-feedback gain K such that the control law guarantees an upper bound of γ ∞ of the closedloop transfer function, while the closed-loop is stable. Note, all the above presented BRL structures applied in the control law synthesis lead to bilinear matrix inequalities and have to be linearized.

Theorem 1. System Eqs. (1) and (2) under control Eq. (3)
is stable with quadratic performance γ ∞ , if there exist a positive definite symmetric matrix R ∈ I R n × n , a matrix Y ∈ I R r × n , and a positive scalar γ ∞ ∈ I R such that  When these inequalities are satisfied, the control law gain matrix is given as Proof. Since P is positive definite, the transform matrix T ∞ can be defined as follows: Then, premultiplying the left side of Eq. (35) and postmultiplying the right side of Eq. (35) by  When these inequalities are satisfied, the control law gain matrix is given as Proof. Considering that Q is positive definite, the transform matrix T ∘ ∞ can be defined as follows: Therefore, premultiplying the left side of Eq. (46) and postmultiplying the right side of Eq. (46) by the matrix T ∘ ∞ gives

Ratio control design
Using the control law Eq. (3), the closed-loop system equations take the form Prescribed by a matrix E ∈ I R p × n , rank E = p ≤ r, it is considered the design constraint Eq. (9) for all nonzero natural numbers i. From Proposition 2, it is clear that such kind of design is a singular task, where Eq. (9) gives which evidently implies Evidently, the equality can be satisfied, as well as the closed-loop system matrix F c = F À GK has to stable (all its eigenvalues are from the unit circle in the complex plane Z).
Lemma 1.The equivalent state-space description of the system Eqs. (1) and (2) under control Eq. (3), in which closed-loop state variables satisfying the condition Eq. (9) is while L ∈ I R r × r is the projection matrix (the orthogonal projector of EG onto the null space N EG [23]) and K°∈ I R r × n is the ratio control gain matrix.
Proof. Premultiplying the left side of Eq. (65) by the identity matrix, it yields which implies the particular solution is the left Moore-Penrose pseudoinverse of EG.
Using the equality Eq. (65), then Eq. (69) can be also written as respectively, where I r ∈ I R p × p is the identity matrix. It is evident that Eq. (74) can be satisfied only if Thus, Eq. (11) implies all solutions of K as follows where K°is an arbitrary matrix with appropriate dimension, and evidently Eq. (76) gives Eq. (68). This concludes the proof. □ Considering the model involving the given ratio constraint on the closed-loop system state variables Eqs. (66)-(68), the design conditions are presented in the following theorems.
Theorem 3. System Eqs. (1) and (2) under the control (3), and satisfying the constraint Eq. (4) is stable with the quadratic performance γ ∞ , if there exist positive definite matrices S, O ∈ I R n × n , a matrix Y°∈ I R r × n , and a positive scalar γ ∞ ∈ I R such that When these inequalities are satisfied, the control law gain matrices are given as where J, L are defined in Eq. (68).
Proof. Substituting Eq. (68) into Eq. (59) gives The ratio control does not exclude a forced regime given by the control law where w(i) ∈ I R m is desired output signal vector and W ∈ I R m × m is the signal gain matrix. Using the static decoupling principle, the conditions to design the signal gain matrix W can be proven.
Lemma 2. If the system Eqs. (1) and (2) is square, which is stabilizable by the control policy Eq. (82) and Ref. [32] rank then the matrix W takes the form where I n ∈ I R n × n is the identity matrix.
Proof. In a steady state, the system equations Eqs. (1) and (2), and the control law Eq. (82) imply where q o , w o are the steady-state values of the vectors q(i), w(i), respectively. Since from Eq. (85), it can be derived that and considering y o = w o , Eq. (87) implies Eq. (84). This concludes the proof. □ Theorem 4. If the closed-loop system state variables satisfy the state constraint Eq. (63), then the common state variable vector q d (i) = Eq(i), q d (i) ∈ I R k attains the steady-state value Proof. Using the control policy Eq. (82), then Since K satisfies Eq. (65), then Eq. (89) implies and it is evident that the tied state variable q d (i) of the closed-loop system in a steady state is proportional to the steady state of the desired signal w o and takes the value Eq. (88). This concludes the proof. □

Illustrative examples
To demonstrate properties of proposed approach, the classical example for a helicopter control [33] is taken, where the discrete-time state-space representation Eqs. (1) and (2) for the sampling period Δt = 0.05s consists of the following parameters The state constraint, defining the ratio control of two state system variables, is specified as It can be easily verified that the closed-loop system matrix takes the format Note that one from the resulting eigenvalue of F c is zero (rank(E) = 1)), because Proposition 2 prescribes this constrained design task as a singular problem. Using the connection between the eigenvector matrix N and M as given by Eq. (17), it is possible to show that this instance is documented also by the structure of M, while Therefore, according to Theorem 4, the constraint given on the states of the system under study is satisfied with zero offset in the autonomous regime and with offset value equal q dw in the forced mode, i.e., The simulation results of the closed-loop system response in the autonomous and forced mode are presented, where Figure 1 is concerned with the system state variables response in the autonomous regime and Figure 2 with the system state variables response in the forced mode. It is evident that the condition Eq. (9) is satisfied at all time instant, except initial time instant in the above given way (see the time response of the additive of variable, which is included as q d (i) in the figures).
For comparison, an example is given for default design of state feedback gain matrix using BRL structure of LMIs. Solving Eqs. (54) and (55), the task is feasible with the Lyapunov matrix variables  the closed-loop system matrix takes the form To apply in the forced mode, the signal gain matrix W is now computed by using Eq. (84) as follows: W ¼ À0:8296 0:9567 À2:2360 2:4922 The simulation results of the nominal closed-loop system response are illustrated in Figures 3  and 4, where Figure 3 is concerned with the system state variables response in the autonomous regime and Figure 4 with the system state variables response in the forced mode.
Since these two control structures are of interest in the context of full-state control design, matching the presented results, it is evident that the system dynamics in both cases are comparable.

Concluding Remarks
In this chapter, an extended method is presented, based on the classical memoryless feedback H ∞ control principle of discrete-time systems, if the ratio control is reformulated by an equality constraint setting on associated state variables. The asymptotic stability of the control scheme is guaranteed in the sense of the enhanced representation of BRL, while resulting LMIs are linear with respect to the system state variables, and does not involve products of the Lyapunov matrix and the system matrix parameters, which provides one way of solving the singular LMI problem. Moreover, formulated as a stabilization problem with the full-state feedback controller, the control gain matrix takes no special structure. The formulation allows to find a solution without restrictive assumptions and additional specifications on the design parameters. It is clear from Theorem 4 that the control law strictly solves the problem even in the unforced mode.