Lizette Zietsman

Control, estimation and optimization of energy efficient buildings

Commercial buildings are responsible for a significant fraction of the energy consumption and greenhouse gas emissions in the U.S. and worldwide. Consequently, the design, optimization and control of energy efficient buildings can have a tremendous impact on energy cost and greenhouse gas emission. Buildings are complex, multi-scale in time and space, multi-physics and highly uncertain dynamic systems with wide varieties of disturbances. Recent results have shown that by considering the whole building as an integrated system and applying modern estimation and control techniques to this system, one can achieve greater efficiencies than obtained by optimizing individual building components such as lighting and HVAC. We consider estimation and control for a distributed parameter model of a multi-room building. In particular, we show that distributed parameter control theory, coupled with high performance computing, can provide insight and computational algorithms for the optimal placement of sensors and actuators to maximize observability and controllability. Numerical examples are provided to illustrate the approach. We also discuss the problems of design and optimization (for energy and CO2 reduction) and control (both local and supervisory) of whole buildings and demonstrate how sensitivities can be used to address these problems.

Mesh Independence of Kleinman-Newton Iterations for Riccati Equations in Hilbert Space

In this paper we consider the convergence of the infinite dimensional version of the Kleinman-Newton algorithm for solving the algebraic Riccati operator equation associated with the linear quadratic regulator problem in a Hilbert space. We establish mesh independence for this algorithm and apply the result to systems governed by delay equations. Numerical examples are presented to illustrate the results.

On using LQG performance metrics for sensor placement

We discuss four metrics for determining sensor placement in energy efficient building design. These include the norm of the observer gain, the trace of the observer Riccati solution, the distance to the nearest unobservable system, and the linear quadratic Gaussian (LQG) cost from given initial state and state estimates. These metrics have different computational complexity, but all lead to the same optimal sensor location in this study where a single room model is considered.

Linear feedback control of a von Kármán street by cylinder rotation

This paper considers the problem of controlling a von Kármán vortex street (periodic shedding) behind a circular cylinder using cylinder rotation as the actuation. The approach is to linearize the Navier-Stokes equations about the desired (unstable) steady-state flow and design the control for the regulator problem using distributed parameter control theory. The Oseen equations are discretized using finite element methods and the resulting LQR control problem requires the solution to algebraic Riccati equations with very high rank. The feedback gains are computed using model reduction in a “control-then-reduce” framework. Model reduction is used to efficiently solve both Chandrasekhar and Lyapunov equations. The reduced Chandrasekhar equations are used to produce a stable initial guess for a Kleinman-Newton iteration. The high-rank Lyapunov equations associated with Kleinman-Newton iterations are solved by applying a novel model reduction strategy. This “control-then-reduce” methodology has a significant computational cost, but does not suffer many of the “reduce-then-control” setbacks, such as ensuring the unknown feedback functional gains are well represented in the reduced-basis. Numerical results for a 2-D cylinder wake problem at a Reynolds number of 100 demonstrate that this approach works when perturbations from the steady-state solution are small enough. When this feedback control is applied to a flow where vortex shedding has already occurred, the feedback control in the nonlinear problem stabilizes a nontrivial limit cycle. This limit cycle does have reduced lift forces and showcases the promise of the linear feedback control approach.

On strong convergence of feedback operators for non-normal distributed parameter systems

We consider the question of strong convergence of the functional gains for LQR/LQG control laws of delay systems. The feedback operators for such systems can be defined in terms of solutions of operator Riccati equations. It has been known for some time that dual convergence is sufficient for strong convergence of the corresponding feedback operators. In this paper, we conjecture that dual convergence is also necessary. Numerical examples are presented to illustrate this point. Finally, we close with a specific conjecture and discuss some previous results along this line.

Approximating parabolic boundary control problems with delayed actuator dynamics

In this paper we consider a control problem for the convection diffusion equation and investigate the impact of including actuator dynamics with delays. The problem is motivated by applications to control of energy efficient buildings where actuation is provided by a HVAC system. The basic model is governed by a parabolic partial differential equation (PDE) with boundary inputs. The boundary inputs are assumed to be the output of an actuator governed by a delay differential equation. Thus, one augments the PDE with a delay equation model of an actuator with delays. The combined system is described by a coupled delay partial differential equation. We show that under suitable conditions, the coupled delay PDE system is well posed in a standard Hilbert space and we use this corresponding abstract formulation to construct numerical methods for control design. We apply these results to a simple 1D boundary control system to illustrate the ideas and numerical methods.

Computational Challenges in Control of Partial Differential Equations

In this paper we consider the problem of computing optimal feedback control laws for systems of partial differential equations. The problem is motivated by fluid flow control and focuses on computational challenges that need to be addressed before one can solve the complex large scale equations that define the functional gains that yield the controllers. Computing these gains is at least as challenging as the original simulation problem and in some cases the computational complexity can be an order of magnitude greater. Examples are presented to illustrate functional gains, to highlight some of the computational complexity problems and to identify important computational challenges, such as using mesh adaptation.

Optimal sensor design for estimation and optimization of PDE systems

In many situations one is interested in the knowledge of the state of a PDE system on a sub-domain or region in the spatial domain. This type of problem occurs naturally when one is interested in regional control. Also, in cases when full spatial control is the goal and the feedback law has compact spatial support so that the observer need only estimate the local spatial behavior of the system. We consider a class of sensor location problems where the goal is to provide optimal zonal estimation and control for a parabolic distributed parameter system. Numerical examples are provided to illustrate the approach. We observed that in many cases the optimization problem is mesh independent and lends itself to parallel and multi-grid optimization approaches.

On the Inclusion of Actuator Dynamics in Boundary Control of Distributed Parameter Systems

Abstract The problem of boundary control in systems governed by partial differential equations often leads to abstract control systems of the form where A generates a C 0 -semigroup, B is an unbounded operator and the system is defined in a very weak sense. Here N (·) is a nonlinear term and w p ( t ) represents a disturbance to the plant. In this setting the unboundedness of the operator B can lead to theoretical and computational challenges. However, in most practical settings the input at the boundary v ( t ) is typically the output of a dynamic “actuator” and the inclusion of actuator dynamics is a more realistic representation of the system. Although the inclusion of actuator dynamics can bring additional complexity to the corresponding control problem, in some cases the formulation of the control system as a composite system is essential. Moreover, in some cases including the actuator dynamics can produce theoretical and computational advantages that can be exploited when introducing approximations. In this paper we discuss various formulations of boundary control problems with actuator dynamics and suggest an alternate approach to formulating certain boundary control problems so that the resulting composite system is well-posed. We apply these results to boundary control of parabolic systems to illustrate the ideas and present numerical results.

Upwind approximations and mesh independence for LQR control of convection diffusion equations

The development of practical computational schemes for optimization and control of non-normal distributed parameter systems requires that one builds certain computational efficiencies (such as mesh independence) into the approximation scheme. We consider some numerical issues concerning the application of Kleinman-Newton algorithms to discretizations of infinite dimensional Riccati equations that arise in control of PDE systems. We show that dual convergence and compactness play central roles in both convergence and mesh independence and we present numerical results to illustrate the theory.