Introduction

In this chapter, several examples of how the theory of complex-valued matrix derivatives can be used as an important tool to solve research problems taken from signal processing and communications. The developed theory can be used to solve problems in areas where the unknown matrices are complex-valued matrices. Examples of such areas are signal processing and communications. Often in these areas, the objective function is a real-valued function that depends on a continuous complex-valued matrix and its complex conjugate. In Hjørungnes and Ramstad (1999) and Hjørungnes (2000), matrix derivatives were used to optimize filter banks used for source coding. The book by Vaidyanathan et al. (2010) contains material on how to optimize communication systems by means of complex-valued derivatives. Complex-valued derivatives were applied to find the Cramer-Rao lower bound for complex-valued parameters in van den Bos (1994b); and Jagannatham and Rao (2004)

The rest of this chapter is organized as follows: Section 7.2 presents a problem from signal processing on how to find the derivative and the Hessian of a real-valued function that depends on the magnitude of the Fourier transform of the complex-valued argument vector. In Section 7.3, an example from signal processing is studied in which the sums of the squared absolute values of the off-diagonal elements in a covariance matrix are minimized. This problem of minimizing the off-diagonal elements has applications in blind carrier frequency offset (CFO) estimation.

Introduction

In this chapter, most of the notation used in this book will be introduced. It is not assumed that the reader is familiar with topics such as Kronecker product, Hadamard product, or vectorization operator. Therefore, this chapter defines these concepts and gives some of their properties. The current chapter also provides background material for matrix manipulations that will be used later in the book. However, it contains just the minimum of material that will be used later because many excellent books in linear algebra are available for the reader to consult (Gantmacher 1959a-1959b; Horn & Johnson 1985; Strang 1988; Magnus & Neudecker 1988; Golub & van Loan 1989; Horn & Johnson 1991; Lütkepohl 1996; Harville 1997; Bernstein 2005).

This chapter is organized as follows: Section 2.2 introduces the basic notation and classification used for complex-valued variables and functions. A discussion of the differences between analytic and non-analytic functions is presented in Section 2.3. Basic matrix-related definitions are provided in Section 2.4. Several results involving matrix manipulations used in later chapters are found in Section 2.5. Section 2.6 offers exercises related to the material included in this chapter. Theoretical derivations and computer programming in MATLAB are topics of these exercises.

Purpose: This volume is devoted to the study of feedback control of so-called linear plant/nonlinear instrumentation (LPNI) systems. Such systems appear naturally in situations where the plant can be viewed as linear but the instrumentation, that is, actuators and sensors, can not. For instance, when a feedback system operates effectively and maintains the plant close to a desired operating point, the plant may be linearized, but the instrumentation may not, because to counteract large perturbations or to track large reference signals, the actuator may saturate and the nonlinearities in sensors, for example, quantization and dead zones, may be activated.

The problems of stability and oscillations in LPNI systems have been studied for a long time. Indeed, the theory of absolute stability and the harmonic balance method are among the best known topics of control theory. More recent literature has also addressed LPNI scenarios, largely from the point of view of stability and antiwindup. However, the problems of performance analysis and design, for example, reference tracking and disturbance rejection, have not been investigated in sufficient detail. This volume is intended to contribute to this end by providing methods for designing linear controllers that ensure the desired performance of closed loop LPNI systems.

The methods developed in this work are similar to the usual linear system techniques, for example, root locus, LQR, and LQG, modified appropriately to account for instrumentation nonlinearities. Therefore, we refer to these methods as quasilinear and to the resulting area of control as quasilinear control.

Motivation: Analysis of disturbance rejection is intended to quantify the standard deviation of the output in closed loop LPNI systems. Unlike the linear case, the usual Lyapunov equation approach cannot be used for this purpose. Therefore, the first goal of this chapter is to provide a quasilinear method applicable to systems with nonlinear instrumentation. While some results in this direction for SISO systems have been described in Chapter 2, we treat here the MIMO case.

Also, we address the problem of fundamental limitations on achievable disturbance rejection in system with saturating actuators. As it is well known, in linear systems with minimum-phase plants, disturbances can be attenuated to any desired level. Clearly, this can not be the case in systems with saturating actuators. Therefore, the second goal of this chapter is to quantify fundamental limitations on disturbance rejection in closed loop LPNI systems and characterize tradeoffs between the authority of the actuators and the achievable disturbance attenuation.

In addition, we present a method for modeling actuators with rate saturation and show that disturbance rejection analysis in LPNI systems with such actuators can be reduced to that with amplitude saturation.

Overview: A quasilinear method for output standard deviation evaluation in closed loop MIMOLPNI systems is developed. The method consists of a Lyapunov equation coupled with transcendental equations defining the quasilinear gains of actuators and sensors. An algorithm for solving these equations with any desired accuracy is provided.

Motivation: The nonlinearities in sensors and actuators lead to performance degradation in LPNI systems as compared with linear ones. In this situation, two questions arise: (1) How can nonlinear instrumentation be selected so that the degradation is no more than a given bound? (2) Is it possible to modify the controller so that for a given nonlinear instrumentation, the linear performance is recovered? These questions motivate the subject matter of this chapter. Specifically, the first question leads to a method of partial performance recovery and the second to a complete performance recovery technique.

Overview: We develop techniques for both partial and complete performance recovery. Specifically, we show that the solution of the former is given by a Nyquist stability criterion-type technique (Section 7.1), while the latter is provided by the so-called controller boosting approach (Section 7.2).

Partial Performance Recovery

Scenario

In practice, control systems are often designed using linear techniques. In reality, control systems often (or, perhaps, always) include saturating actuators. The question arises: How large should the level of saturation be so that the performance predicted by linear design does not degrade too much? In this section, this question is addressed in the framework of the disturbance rejection problem.

This volume has extended major ideas and results of linear control theory to systems with nonlinear actuators and sensors. Namely, the linear frequency domain methods (based on the frequency of harmonic inputs) are extended to the quasilinear frequency domain (based on the bandwidth of random inputs). The linear time domain methods are extended to quasilinear ones by introducing tracking quality indicators (instead of overshoot and settling time) and by modifying the root locus technique (e.g., the saturated root locus). The LQR/LQG methodology has been extended to SLQR/SLQG (where the “S” stands for “saturated”). Although not included in this volume, the H∞ and LMI techniques have also been extended to systems with nonlinearities in sensors and actuators.

In addition, new problems, specific for LPNI systems, have been formulated and solved. These include Instrumented LQR/LQG, where the optimal controller and instrumentation are synthesized simultaneously, and the problem of performance recovery, where the performance losses due nonlinear instrumentation are either contained to a desired level or eliminated altogether.

What made these results possible?

1. The method of stochastic linearization provided a tool for reducing the nonlinear systems addressed in this volume to quasilinear ones.

2. The methods of linear control theory could be applied, with some modifications, to the quasilinear systems mentioned above.

3. The resulting relationships and techniques, as it turns out, are quite similar to those of linear control theory.

Motivation: This chapter is intended to introduce the class of systems addressed in this volume – the so-called Linear Plant/Nonlinear Instrumentation (LPNI) systems – and to characterize the control methodology developed in this book – Quasilinear Control (QLC).

Overview: After introducing the notions of LPNI systems and QLC and listing the problems addressed, the main technique of this book – the method of stochastic linearization – is briefly described and compared with the usual, Jacobian, linearization. In the framework of this comparison, it is shown that the former provides a more accurate description of LPNI systems than the latter, and the controllers designed using the QLC result, generically, yield better performance than those designed using linear control (LC). Finally, the content of the book is outlined.

Linear Plant/Nonlinear Instrumentation Systems and Quasilinear Control

Every control system contains nonlinear instrumentation – actuators and sensors. Indeed, the actuators are ubiquitously saturating; the sensors are often quantized; deadzone, friction, hysteresis, and so on are also encountered in actuator and sensor behavior.

Typically, the plants in control systems are nonlinear as well. However, if a control system operates effectively, that is, maintains its operation in a desired regime, the plant may be linearized and viewed as locally linear. The instrumentation, however, can not: to reject large disturbances, to respond to initial conditions sufficiently far away from the operating point, or to track large changes in reference signals – all may activate essential nonlinearities in actuators and sensors, resulting in fundamentally nonlinear behavior.

Motivation: In designing linear feedback systems, tracking quality specifications are typically mapped into admissible domains in the complex plane and then the root locus technique is used to synthesize a controller that places closed loop poles in the desired locations. This methodology motivates the first goal of this chapter: to generalize the notion of admissible domains and the root locus technique to LPNI systems. Accomplishing this goal leads to a quasilinear method for PID controller design.

Overview: A technique for mapping the random reference tracking quality indicators, introduced in Chapter 3, into admissible domains on the complex plane is introduced (Section 5.1). Next, the root locus method for systems with saturating actuators is developed (S-root locus, Section 5.2). It turns out that the S-root locus is a subset of the usual root locus typically terminating prior to the open loop zeros. A method for calculating these termination points is provided. In addition, we equip the S-root locus with the so-called truncation points, which define its segments where tracking without amplitude truncation is possible.

Admissible Pole Locations for Random Reference Tracking

Scenario

In linear Control Theory, admissible domains for tracking deterministic references are defined using the classical “quality indicators,” that is, overshoot and settling, rise, and delay times. In this section, we develop a similar approach for tracking random references.

Motivation: The LQR and LQG techniques are widely used for designing disturbance rejection controllers for linear systems. In this chapter, quasilinear versions of these techniques are developed and extended to simultaneous design of controllers and instrumentation, that is, actuators and sensors.

Overview: We begin with developing the SLQR/SLQG technique, where the “S” stands for “saturated” (Section 6.1). Then, we extend LQR/LQG to the so-called ILQR/ILQG (where the “I” stands for “instrumented”), which is intended for simultaneous design of controllers, sensors, and actuators (Section 6.2). For each of these methods, we show that the solution is given by the usual Lyapunov and Ricatti equations coupled with additional trancendental equations, which account for the Lagrange multipliers of the associated optimization problems and the quasilinear gains. Bisection algorithms for solving these coupled equations are provided.

Saturated LQR/LQG

Scenario

The LQR/LQG design methodology is one of the main state space techniques for designing linear controllers for linear plants, assuming that the actuators and sensors are also linear. Clearly, this methodology is not applicable to nonlinear systems. Is an extension to LPNI systems possible? A positive answer to this question is provided in this section for systems with saturating actuators. The resulting technique is referred to as SLQR/SLQG, where the “S” stands for “saturated.” Extensions to other nonlinearities and to nonlinearities in actuators and sensors simultaneously are also possible using the approach presented here.

Motivation: This chapter is intended to quantify steady state errors and transient performance of LPNI systems in the problem of tracking reference signals.

For linear systems, the steady state error in tracking deterministic references is characterized by the notion of system type, defined by the poles of the loop gain at the origin. This definition is not applicable to LPNI systems since, if the actuator saturates, controller poles at the origin play a different role than those of the plant and, thus, the loop gain does not define the system type. This motivates the first goal of this chapter, which is to introduce the notion of system type for LPNI systems.

Also in contrast with the linear case, LPNI systems with saturating actuators are not capable of tracking steps with arbitrary amplitudes. This motivates the second goal of this chapter: to introduce and quantify the notion of trackable domains, that is, the sets of step (or ramp, or parabolic) input magnitudes that can, in fact, be tracked by an LPNI system.

As far as the transients are concerned, the usual step tracking measures, such as overshoot, settling time, and so on, are clearly not appropriate for random references. The variance of tracking errors, as it turns out, is not a good measure either, since for the same variance, the nature of tracking errors can be qualitatively different.

Introduction

In this chapter, the theory and a resulting indirect method of trajectory optimization are derived and illustrated. In an indirect method, an optimal trajectory is determined by satisfying a set of necessary conditions (NC), and sufficient conditions (SC) if available. By contrast, a direct method uses the cost itself to determine an optimal solution.

Even when a direct method is used, these conditions are useful to determine whether the solution satisfies the NC for an optimal solution. If it does not, it is not an optimal solution. As an example, the best two-impulse solution obtained by a direct method is not the optimal solution if the NC indicate that three impulses are required. Thus, post-processing a direct solution using the NC (and SC if available) is essential to verify optimality.

Optimal Control, a generalization of the calculus of variations, is used to derive a set of necessary conditions for an optimal trajectory. The primer vector is a term coined by D. F. Lawden in his pioneering work in optimal trajectories. [This terminology is explained after Equation (2.24).] First-order necessary conditions for both impulsive and continuous-thrust trajectories can be expressed in terms of the primer vector. For impulsive trajectories, the primer vector determines the times and positions of the thrust impulses that minimize the propellant cost. For continuous thrust trajectories, both the optimal thrust direction and the optimal thrust magnitude as functions of time are determined by the primer vector.