Synthesis Conditions for LPV Controller with Input Covariance Constraints

Abstract In this paper, novel synthesis conditions for state-feedback Linear Parameter Varying (LPV) controller with Input Covariance Constraints (ICC) are developed. The synthesis conditions achieve the following design requirements 1) some constraints need to be satisfied on the control energy and 2) optimizing the performance outputs for the entire parameter space of the LPV system. These conditions are formulated as convex optimization problem with Parameterized Linear Matrix Inequalities (PLMIs) constraints. The effectiveness of the proposed approach is illustrated through numerical examples.


Introduction
Linear Parameter Varying (LPV) control techniques witnessed great interest in the past three decades [1,2]. LPV systems can be divided into two main categories, gridding and polytopic LPV models [3]. The gridding LPV control is based on linearizations at speci c local points, therefore, there is no guarantee of stability and/or performance for the entire range of the parameter space [4]. On the other hand, polytopic LPV systems provide this guarantee in addition to the simplicity of controller design [5]. Lyapunov approach is used to analyze stability for polytopic LPV systems [6][7][8].
Since all actuators require limited amount of the available power, practical control systems have to operate below a speci ed constraints. In the case of designing controllers using classical design methods, the limits of the operation of the actuators are not considered through the design stage. Therefore, the controller could command the actuator to operate beyond its allowable range that may damage the actuator. Therefore, it is always required to keep the actuators working in their linear range and satisfying their constraints. If the control signal exceeds the constraints, the closed-loop system will operate beyond its linear range and even stability may no longer guaranteed. This control problem can be solved via Input Covariance Constraint (ICC) controller [9,10].
The ICC control problem is a problem that optimize output performance such that constraints on the control input are satis ed. In other words, it guarantees the actuators work within their linear range of operation.
The ICC control problem can be interpreted in two ways, deterministic and stochastic interpretation. In the deterministic interpretation, the external disturbance inputs are assumed to belong to a bounded energy L set. By de ning these inputs this way, the ICC control problem is the problem of minimizing the weighted sum of worst-case peak values on the performance outputs subject to the constraints on the worst-case peak values of the control input. On the other hand, these external inputs can be considered uncorrelated zero-mean white noises with specied intensity. Thus, the ICC control problem minimizes the weighted output performance subject to the control input covariance constraints.
The ICC control problem has been studied earlier for Linear Time Invariant (LTI) systems in [11]. Then, the problem had been revisited in [12] with Linear Matrix Inequalities (LMIs) approach. Tensegrity simplex structure is studied in [13] with ICC controller to satisfy torque actuator constraints. In [14], electric throttle is investigated with experimental validation of ICC controller. The constrained LQ control problem is studied in [15] that minimizes the control energy while satisfying Output Covariance Constraints (OCC).
As far as the author's know, this control problem has never been studied for LPV systems before. Therefore, the objective of this paper is to extend the ICC control problem presented in [12] for Linear Time-Invariant (LTI) to handle LPV systems. More speci cally, we have developed new synthesis conditions to design state-feedback LPV controller to asymptotically stabilize the closed-loop sys-tems while satisfying constraints on control signals. These synthesis conditions are formulated as convex optimization problem with Parametrized Linear Matrix Inequalities (PLMIs) constraints. Thus, the developed conditions have in nite constraints. Therefore, matrix coe cient check approach has been used to relax these conditions into a nite set of numerically tractable LMIs [16].
The paper is organized as follows. Section 2 presents preliminaries of the LPV systems and signals norm. Section 3 presents the synthesis conditions of the ICC LPV controller. Numerical examples are given in Section 4 to demonstrate the bene t of the developed conditions. Closing conclusions are given in section 5 Standard notation is used in this paper. R denotes the set of real numbers. The symbol "T" denotes transpose of a matrix. The matrices In and n denote an identity and zero matrices of size n × n, respectively. The positive deniteness of a matrix A is denoted A > . The sum of the diagonal elements of a matrix A is referred as trace(A). The transposition of the o -diagonal matrix block is referred by " * ". Other notations will be illustrated in due course.

Preliminaries
Consider the following linear parameter varying system : where q is the number of scheduling parameters. The scheduling parameters are assumed to be bounded such that, Moreover, the rate of change of the scheduling parameters are assumed to be bounded as well.
The LPV system in (1) can be converted into polytopic coordinate using the change of variables, where µ := (µ (t), µ (t)) ∈ Λ and where the subscript " " denotes the number of vertices. Following the procedure presented in [17,18], the derivatives of the varying parameters can assume values bounded by convex polytope Ω. Thus, the system in (1) can be expressed in the polytopic coordinates (in terms of µ(t)) aṡ The L de nition of a signal is given by while the L∞ is de ned then, the L -to-L∞ gain for Linear Time Invariant (LTI) systems is de ned by where T is the transfer function from the disturbance input w(t) to performance output y(t).

Input Covariance Constraints (ICC)
The control objective is to design gain-scheduling statefeedback controllers of the form in order to: • stabilize the closed-loop system.
• satisfy constraints on input covariances de ned by U.
• minimize the output performance in terms of the Hnorm of the closed-loop system from the disturbance input Augmenting the open-loop system (1) with the controller (8), the following closed-loop system matrices are obtained, with A(µ) := A(µ) + Bu(µ)K(µ).
Assume A(µ) is stable for any pair (µ,μ) ∈ Λ × Ω, then there is a parameter-dependent positive-de nite matrix P(µ) = P(µ) T > satisfying the following inequalitẏ In other words, suppose there is a continuously differentiable parameter-dependent matrix P(µ) = P(µ) > that solves the parameter-dependent controllability Gramian of the LPV system, (12) that implies P(µ) > P(µ). In other words, P(µ) is a parameter-dependent covariance matrix and the control covariance can be de ned, where Φ k is the k-th row of an identity matrix of size nu ×nu associated with the k-th control channel.
Hence, the controller should exponentially stabilizes the closed-loop system and minimizes the upper bound of the ICC cost de ned by, Since the cost is parameter-dependent also, we introduce additional semi-de nite (parameter-independent) matrix W such that Therefore, minimizing the trace(W) corresponding to minimizing the upper bound of the ICC cost, More speci cally, the goal is to nd a gain-scheduling state-feedback controller to minimize (16) subject to the constraints (13). Theorem 1. Given input covariance constraints U k for k = , , · · · , nu, and a su ciently small scalar ϵ > . If there exist continuously di erentiable parameter-dependent matrix < P(µ) = P(µ) T ∈ R n×n , G(µ) ∈ R n×n , Z(µ) ∈ R nu×n , and parameter-independent matrix W = W T ∈ R ny×ny that solve the following optimization problem then the gain-scheduling controller stabilizes the closed-loop system for any (µ,μ) ∈ Λ × Ω, with the ICC cost bounded by
Following similar steps, the PLMIs (19) can be written as U k > Φ k K(µ)P(µ)K(µ) T Φ T k for k = , , · · · , nu, that implies the control covariance bounded by U k . Some remarks are due for Theorem 1 in sequential order.

Corollary 3. Given input covariance constraints U k for k =
, , · · · , nu, and a su ciently small scalar ϵ > . If there exist continuously di erentiable parameter-dependent matrix < P(µ) = P(µ) T ∈ R n×n , parameter-independent matrices G ∈ R n×n , Z ∈ R nu×n , and W = W T ∈ R ny×ny that solve the following optimization problem then the robust ICC controller stabilizes the closed-loop system with guaranteed ICC cost bounded by where Φ k is (suitably chosen) input channel selection matrix and ψr := A(µ)G + B(µ)Z.

Remark 4. The synthesis conditions of Theorem 1 are parametrized with µ(t) (polytopic vertices) instead of θ(t) since a simple change of variables can be used to convert the original LPV model into the polytopic domain.
Remark 5. The conditions of Theorem 1 are expressed as Parametrized Linear Matrix Inequalities (PLMIs). Hence, PLMIs can be interpreted as special type of convex optimization with in nite constraints. Therefore, relaxation methods need to be applied in order to solve this optimization problem. In this work, matrix coe cients check relaxation approach using Pólya's theorem [16] is utilized to obtain the optimal LPV controller. (17)
Then, Theorem 1 is used to design LPV controllers with various constraints on the control input. Table 1 shows the number of LMI (optimization) variables and LMI constraints for Theorem 1 associated with Example 1.
Three cases have been studied with input constraints, U = . , and . , respectively. For the rst case,  Table 2: ICC cost and Control covariance at the system vertices for Example 1 (ϵ = − ). U = . is used in Theorem 1 to design the optimal LPV controller satisfying this control constraints. Table 2 illustrates the design results at the vertices of the polytope with the vertex controller,

# of LMI variables # of LMI rows Theorem
] . (28) It is clear from Table 2 that the utilized control energy at the two vertices are . and . which satisfy the constraint U = . . With this constraint, the optimal ICC costs are . and . at the vertices. Note that trace(W) = .
is the upper bound for the ICC cost. This bound is very tight demonstrating optimality of the design.
For the case U = , the utilized control energy at the vertices are . and . with upper bound of the ICC cost is 0.0578. Similarly, for the case U = . , the utilized control energy at the vertices are . and .
with upper bound of the ICC cost is 0.0575. Figure 1 illustrates the relationship between the optimized output performance versus the control constraints. As expected, the output performance is getting better as the constraints on the control relaxed and vise versa.
The worst case control response for U = is shown in Fig. 2 for both vertices demonstrating the control input constraint is totally respected. This response illustrating that the control resources are optimally utilized, neither commanding the actuator higher than the speci ed constraint nor providing low actuation signal. . Example: 2 Consider the modi ed version of the system presented in [11]: A(θ(t)) = with θ(t) ∈ [ , ] andθ(t) ∈ [− , ]. After converting the system into polytopic domain, Theorem 1 is used to synthesize LPV controller with ICC constraints. Table 4 shows the number of optimization variables and LMI constraints associated with Theorem 1 for this example. Similarly, three cases have been studied with constraints U = . , , and . Table 3 presents the design results in terms of vertex controllers and the achieved output performance for each individual case. It is clear to see the control energy satisfying the speci ed constraints. Figure 3 demonstrates the relationship between the control constraints and the achieved J ICC .