Get Complete Project Material File(s) Now! »

## Traditional methods for controller design

Traditional methods for designing stabilising and optimal H1 controllers for LTI MIMO plants with time-delays are grounded in the Riccati equation and LMI framework (see (Fridman, 2014) and references therein) using sufficient conditions for stability. In general, controllers designed by these methods are not structured and their dimension is equal or larger than the order of the plant (full order controllers). The classical approach for control design based on matrix inequalities with an unknown output-feedback static controller gain matrix gives rise to a bilinear (non-convex) optimisation problem. Several attempts to solve this optimisation problem can be found in the literature (Chen and Zheng, 2006; Fridman and Shaked, 2002; Li and de Souza, 1997; Zeng et al., 2015; Zhang et al., 2005). For example, in (Barreau et al., 2018), an iterative LMI procedure which takes advantage of the elimination lemma was introduced to solve the bilinear optimisation problem. Intuitively, the problem does not become simpler for the design of robust and stabilising structured (for decentralised, overlapping, or distributed controllers) or fixed-order controllers for systems with multiple delays at input, output, and state. Therefore, this thesis focuses on using frequency domain-based direct optimisation techniques of (Michiels, 2011) and (Gumussoy and Michiels, 2011) grounded in necessary and sufficient conditions for stability which are adequate for designing structured (decentralised, distributed, PID, and overlapping) and lower-order (or fixed-order) controllers. However, using this approach implies that only LTI systems with constant time delays can be considered for control design. Then, some nonlinear and time-varying properties of the (linearised) systems may be treated later, on the basis of the resulting LTI controller, using techniques from robust control theory (which are generally conservative). The objective functions used for optimisation of the controller parameters in the frequency domain are described in the following subsections.

### Robust spectral abscissa optimisation

In this thesis, we focus on (strong) exponential stability of LTI systems with constant time-delays. The notion of exponential stability (of the null solution) is defined as follows for (1.1) when the inputs are zero (Michiels and Niculescu, 2014).

Definition 1.5.1 (Exponential stability) The null solution of (1.1), when u 0 and w 0, is exponentially stable if and only if there exist constants 1 > 0 and 2 > 0 such that,8 2 X, 8 t 0, ||(xp())(t)|| 1e−2t||||s, where || · ||s is the supremum norm, ||||s = sup˜2[−max,0] ||(˜)||2. However, plant (1.1) need not be stable and we would like to design a controller that guarantees the exponential stability of the closed-loop system (1.5). For this purpose, the frequency domain-based direct optimisation technique is adopted for the reasons mentioned in Section 1.5.1. The spectral abscissa of the closed-loop system (1.5) with w 0 is defined as follows.

#### Strong H1 norm optimisation

The H1 norm was introduced in the 1970s and early 1980s, see (Helton, 1978; Tannenbaum, 1980; Zames, 1981). Subsequently, the H1 methods were developed and are now routinely used for control design. The H1 norm may be used for designing systems with optimal (energy) gain from input disturbances w to some output signal z. Additionally, the robustness of stability for systems with parametric uncertainties can be recast in terms of the H1 norm. In this thesis, the system performance levels are expressed in terms of the H1 norm. The transfer function matrix from w to z of the system represented by (1.5) is given by Under assumption of internal stability, the H1 norm of the transfer function matrix given in (1.12) can be expressed as ||G(j!, ¯p; ¯ )||H1 = sup !2R 1(G(j!, ¯p; ¯ )),

**Exploiting network structure of systems**

We particularise the approach (for generic systems) presented in Section 2.2 for the case of designing a decentralised (or overlapping) controller for a special class of systems. These systems have some network structure, consisting of identical subsystems to be controlled by identical local controllers, see Fig. 2.1b. We consider the case where a system of the form (1.1) is composed of subsystems of the form (1.20) and (1.21). We assume that each subsystem can be controlled using a fixed-order LTI feedback controller of the form 8< x˙ ci(t) = Aˆcxci(t) + ˆBcyi(t), ui(t) = ˆ Ccxci(t) + ˆDcyi(t), i = 1, …, n.

**Extension to a scalable algorithm**

From Theorem 2.3.1, the design of stabilising decentralised controllers can be recast as the simultaneous stabilisation problem of finding controller parameters such that (2.17), with ¯ wi 0, is exponentially stable 8 i = 1, . . . , n. For some typologies, the eigenvalues corresponding to their network adjacency matrices (ai) for any number of nodes (n) are confined to a real interval, i.e ai 2 := [p, q] where p 2 R, q 2 R and p < q. Since the only difference between the systems’ equations in (2.17) lies in parameters ai, a sufficient condition for stability is given by the robust stability of the uncertain system ˆE ˙(t) = Aˆ0 + a ˆB ˆ C (t) + Xm k=1 Aˆk(t − k), (2.45).

subject to interval uncertainty a 2 [p, q], for which necessary and sufficient conditions can be obtained within the real structured pseudospectral framework developed in (Borgioli and Michiels, 2018). The characteristic matrix of (2.45) can be written as ˘M () = ˆE − .

**Table of contents :**

**1 General introduction **

1.1 Context

1.2 Control architectures

1.3 Literature on design of decentralised controllers

1.4 LTI time-delay systems

1.5 Stabilisation and fixed-order controller design

1.5.1 Traditional methods for controller design

1.5.2 Robust spectral abscissa optimisation

1.5.3 Strong H1 norm optimisation

1.6 General problem setting

1.6.1 Systems with network structure

1.6.2 Challenges

1.7 Structure of the thesis

**2 Design of decentralised controllers **

2.1 Introduction

2.2 Design of structurally constrained controllers

2.3 Exploiting network structure of systems

2.3.1 Decoupling for the stabilisation problem

2.3.2 Decoupling for the H1 optimisation problem

2.3.3 Discussion

2.3.4 Generalisations to distributed controllers

2.3.5 Numerical examples

2.4 Extension to a scalable algorithm

2.4.1 Controller design approach

2.4.2 Numerical example

2.5 Conclusions

**3 Decentralised controllers in a network of sampled-data systems **

3.1 Introduction

3.2 MIMO plant and decentralised controllers

3.2.1 Sampled-data decentralised control

3.2.2 A feedback interconnection interpretation

3.3 Stability criterion: generic case

3.4 Controller design

3.4.1 Generic case

3.4.2 Network structure exploitation

3.5 Numerical example

3.6 Conclusions

**4 Application to cooperative adaptive cruise control **

4.1 Introduction

4.2 Vehicle model

4.3 One vehicle look-ahead platoon with CACC

4.4 Stability and performance objectives

4.4.1 Platoon stability: spectral abscissa

4.4.2 Motivation for string stability

4.4.3 Platoon string stability

4.4.4 Platoon stability: pseudospectral abscissa

4.4.5 Investigating a robust string stabilising controller

4.5 Simulation-based studies

4.6 Conclusions

**5 General conclusions **

5.1 Summary

5.2 Future work

A Frequently used network topologies

B Small gain theorem

C Well posedness of the systems considered

D Consensus problem in a ring network topology

E Upper-bound for the operators

**Bibliography**