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## Youla-Kucera state-of-the-art review and applications

This chapter corresponds to a state-of-the-art review that was carried out from the applications of YK and dual YK parametrizations point of view. These parametrizations have been used in different practical control sys-tems, covering controller reconfiguration, identification, Plug&Play (P&P) control, disturbance rejection, adaptive control, and fault tolerant control.

YK parametrization was formulated in the late 70s for obtaining the set of controllers stabilizing a linear plant. This fundamental result of control theory has been used to develop theoretical tools solving many practical control problems. This chapter collects the recent work and classifies them according to the use of YK parametrization or dual YK parametrization or both, providing the latest advances with main successful applications developed in the last two decades in different control fields. A final discussion gives some insights on the future trends in the field.

The rest of the chapter is organized as follows: Section 2.1 presents fundamentals and mathematical basis of YK and dual YK parametriza-tions. Section 2.2 presents the use of the YK parameter Q as a technique of controller reconfiguration for both Linear Time-Invariant (LTI) and Linear Parameter-Variant (LPV) systems. Disturbance and noise rejection con-trol structure based on the YK parameter Q is explained in section 2.3. In section 2.4, the use of dual YK parameter in closed-loop identification is presented. The combination of both YK parameter and dual YK parameter in adaptive control field is detailed in section 2.5. The fault tolerant control scheme based on YK framework is detailed in section 2.6. In section 2.7, the use of YK parametrizations in Plug & Play control and Multi Model Adaptive Control is briefly reviewed. Finally, some challenging issues about the use of YK parametrization are discussed in the last section.

### Origins of YK parametrization

The origin of the YK parameterization is in the 70s, Youla [4,5] and Kucera [6] developed the scientific basis. They proposed a parameterization that provides all linear stabilizing controllers for a given LTI plant in a feedback control loop (see Fig. 2.1). All stabilizing controllers are parametrized based on the transfer function called YK parameter Q, leading to a control form K(Q). The parameter Q(s) is the one guaranteeing the stability. Similarly, its dual theory (also known as the dual YK parametrization) provides all the linear plants stabilized by a given controller. The class of all the plants stabilized by a controller depends on the transfer function called dual YK parameter S, so G(S). This parameter could represent any plant variations. Hence, this useful way of parametrizing either plants, controllers or both is employed to solve many control issues.

Figure 2.1: Feedback control loop block diagram.

According to the control objectives, three main configurations can be targeted:

• Controller parametrization allows stable controller reconfiguration when some change occurs. It is also widely used in disturbance and noise rejection control. A number of successful applications can be found in the last two decades, being the most used approach in different control fields [7].

• Plant parametrization is employed to solve the problem of closed-loop identification. Some successful implementation can be found in Plug & Play control where the dynamics of new sensors or actuators are identified on real-time without system disconnection [8].

• Simultaneous control and plant parametrization provides a new control structure that changes according to new identified dynamics on the plant. This principle is mainly used in fault tolerant control and adaptive control [9].

Let’s consider a Single-Input-Single-Output (SISO) stable plant G(s) connected to a given controller K(s) in a stable feedback loop depicted in fig. 2.1. Closed-loop transfer function CL(s) from reference r to output y is in the following equation:

CL(s) = K(s)G(s)/ 1 + K(s)G(s)

The transfer function from the reference r(s) to the controller output u(s) yields:

Q(s) = K(s) /1 + K(s)G(s)

if Q(s) and G(s) are known, the controller K(s) can be expressed as: K(s) = Q(s) / 1 − G(s)Q(s)

If K(s) is stabilizing G(s), Q(s) is stable and proper. Reciprocally, if Q(s) is stable and proper, it is easy to demonstrate that K(s) stabilizes G(s) using Eq. 2.3.

Thus, stabilizing controllers can be parametrized in terms of the set of all stable proper functions Q(s) for a given plant G(s).

Furthermore, the closed-loop transfer function CL(s) in Eq. 2.1 becomes linear in terms of Q(s) and G(s) which is not the case with K(s).

CL(s) = G(s)Q(s) (2.4)

From those two results (2.3,2.4), the concept of controllers parametriza-tion appeared in [10] applied to scalar only known and stable plants.

Youla et al. [4,5] and Kucera [6] explained simultaneously how the initial idea of controller parametrization can be extended to cover Multi-Inputs-Multi-Outputs (MIMO) plants and that are not necessarily stable. A set of all stabilizing controllers for a given plant is characterized using the so called YK parameter Q(s).

Conversely, a dual concept is proposed by leading the same reasoning in equations (2.2) and (2.3) and use the fact that K(s) and G(s) are com-mutative. The set of all plants stabilized by a given stabilizing controller is characterized using the so called dual YK parameter S(s) [8].

#### Coprime factorization

Both YK and dual YK parametrization are based on the doubly coprime factorization [11, 12] to reduce algebraic complexity in computing Q(s) and S (s).

The plant model and controller matrix transfer functions are factorized as a product of a stable transfer function matrix and a transfer function matrix with a stable inverse with no common unstable zeros as follows:

G = NM−1 = M˜−1N˜ (2.5)

K = UV −1 = V˜−1U˜ (2.6)

These coprime factors are calculated using pole placement technique (see section 3.3.1 for details) to satisfy the following double Bezout identity [13]:

« −N˜ M˜ # » N V# « N V # « −N˜ M˜# » 0 I#

˜ ˜ U = M ˜ ˜ = I 0 (2.7)

V −U M U V −U

**Q-parametrization**

The YK parametrization Q describes how the set of all stabilizing controllers for a given plant G(s) can be characterized from knowing a controller K(s) stabilizing the given plant G(s).

Lemma 1. Let a plant G = M N−1, with N and M coprime and stable, be stabilised by a controller K = U V −1, with U and V coprime and stable. Then the set of all stabilizing controllers for G is given as a function of a stable filter YK parameter Q with appropriate dimensions (see [7]):

K = K(Q) = (U + MQ)(V + NQ)−1

**S-parametrization**

The dual-YK parametrization S describes how the set of all the plants sta-bilized by a given controller K(s) can be characterized from knowing a plant G(s) stabilized by the given controller K(s).

Lemma 2. Let a plant G = N M−1, with N and M coprime and stable, be stabilized by a controller (in positive feedback loop) K = U V −1, with U an V are coprime and stable. Then the set of all the plant stabilized by K

is given as: nG(S) = (N + SV )(M + SU)−1o G = nG(S) = (M˜ + U˜ S)−1(N˜ + V˜ S)o

**(Q,S)-parametrization**

Finally, let’s consider the connection between both parametrizations. The parametrized controller K(Q) described in Eq. 2.3 is connected to the parametrized plant G(S) described in Eq. 2.8. The resulting closed loop is in Fig. 2.2.

Lemma 3. The stability of [G(S), K(Q)] is equivalent to the stability of the positive closed-loop [Q, S] [9].

Hence, this useful linear way of parametrizing either controllers, plants or both is employed to solve many control issues.

**Table of contents :**

**1 Introduction **

1.1 Motivation

1.2 Objectives

1.3 Manuscript organization

1.4 Contribution

1.5 Publication

1.5.1 Journals

1.5.2 Conferences

1.5.3 Patents

**2 Youla-Kucera state-of-the-art review and applications **

2.1 Origins of YK parametrization

2.1.1 Coprime factorization

2.1.2 Q-parametrization

2.1.3 S-parametrization

2.1.4 (Q,S)-parametrization

2.2 Q-based controller reconfiguration

2.2.1 Control implementation

2.2.2 Applications

2.3 Q-based noise rejection and vibration control

2.3.1 Control implementation

2.3.2 Applications

2.4 S-based closed-loop identification

2.4.1 Identification scheme

2.4.2 Applications

2.5 (Q,S)-based adaptive control

2.5.1 Control implementation

2.5.2 Applications

2.6 (Q,S)-based fault tolerant control

2.6.1 Control implementation

2.6.2 Applications

2.7 (Q,S)-P&P and Multi Model Adaptive Control

2.8 YK and Dual-YK Time Evolution

2.9 Discussion

**3 Youla-Kucera parametrization: Theory and principles **

3.1 Introduction

3.2 Single controller limitations

3.2.1 Single multi-criteria controller

3.2.2 Linear controller switching

3.3 YK: definition and stability

3.3.1 Doubly Coprime Factorization

3.3.2 The class of all stabilizing controllers

3.3.3 Controller transition

3.3.4 Youla-Kucera controller stability analysis

3.4 Numerical example

3.5 Conclusion

**4 Transient behavior analysis for YK control structures**

4.1 YK control structure implementations

4.2 Transient behavior

4.2.1 γ placement

4.2.2 Initial and final controllers dynamics

4.2.3 Switching frequency

4.3 Conclusion

**5 Lateral Control**

5.1 Experimental platform description

5.2 Problem description

5.3 Lateral control review

5.4 Motivation on using Youla Kucera

5.5 Control design

5.5.1 Vehicle model

5.5.2 Steering actuator

5.5.3 Lateral controller

5.5.4 Stability Analysis

5.6 Simulation results

5.7 Experimental results

5.8 Conclusion

**6 Longitudinal Control**

6.1 Problem description

6.2 Longitudinal control review

6.3 Motivation on using Youla Kucera

6.4 Control design

6.4.1 Performant controller criteria

6.4.2 Robust controller criteria

6.4.3 Nominal controllers design

6.4.4 Parametrized controller K(Q)

6.5 H∞ controller

6.6 Comparison results

6.6.1 Frequency domain comparison

6.6.2 Temporal domain comparison

6.7 Experimental results

6.8 Conclusion

**7 Discussion**

7.1 Conclusion

7.2 Future works